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{| border="1" cellpadding="4" align="center"
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<wikitex>
!style="background:#efefef;"| *
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{{TODO|t=НЕ ОЧЕНЬ ПОНИМАЮ, ЗАЧЕМ ВООБЩЕ ЭТО УТСВЕРЖДЕНИЕ ТУТ}}
!style="background:#efefef;"| <big>e</big>
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{{Теорема
!style="background:#efefef;"| <big>a</big>
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|statement=
!style="background:#efefef;"| <big>b</big>
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Если $f$ — функция ограниченной вариации ($f \in \bigvee(a, b)$), то ее можно представить в виде разности монотонно неубывающих функций ($f = f_1 - f_2$).
!style="background:#efefef;"| <big>c</big>
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|proof=
!style="background:#efefef;"| <big>d</big>
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Возьмем в качестве $f_1$ функцию $f_1(x) = \bigvee\limits_a^x (f)$, тогда по аддитивности она будет не убывать.
!style="background:#efefef;"| <big>f</big>
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Определим как $f_2$ функцию $f_2(x) = f_1(x) - f(x)$. Докажем, что она монотонно не убывает.
|-
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$a < x_1 < x_2 < b$. Надо доказать, что $f_1(x_1) - f(x_1) \le f_1(x_2) - f(x_2)$, или что $f(x_2) - f(x_1) \le f_1(x_2) - f_1(x_1) = \bigvee\limits_{x_1}^{x_2} (f)$ (используем утверждение 1).
!style="background:#efefef;"| <big>e</big>
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Но действительно $f(x_2) - f(x_1) \le | f(x_2) - f(x_1) | \le \bigvee\limits_{x_1}^{x_2} (f)$, ч. т. д.
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
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}}
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big>
 
|}
 
  
{| border="1" cellpadding="4" align="center"
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</wikitex>
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!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>c</big> || <big>f</big> || <big>b</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>b</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>e</big> || <big>b</big> || <big>a</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>e</big> || <big>a</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>a</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>c</big> || <big>e</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>a</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>c</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>e</big> || <big>d</big> || <big>a</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>e</big> || <big>d</big> || <big>b</big> || <big>c</big> || <big>a</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>a</big> || <big>f</big> || <big>e</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>e</big> || <big>d</big> || <big>b</big> || <big>c</big> || <big>a</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>b</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>e</big> || <big>d</big> || <big>a</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>e</big> || <big>f</big> || <big>b</big> || <big>a</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>a</big> || <big>d</big> || <big>f</big> || <big>e</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>e</big> || <big>b</big> || <big>c</big> || <big>a</big>
 
|}
 
 
 
{| border="1" cellpadding="4" align="center"
 
!style="background:#efefef;"| *
 
!style="background:#efefef;"| <big>e</big>
 
!style="background:#efefef;"| <big>a</big>
 
!style="background:#efefef;"| <big>b</big>
 
!style="background:#efefef;"| <big>c</big>
 
!style="background:#efefef;"| <big>d</big>
 
!style="background:#efefef;"| <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>e</big>
 
| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big>
 
|-
 
!style="background:#efefef;"| <big>a</big>
 
| <big>a</big> || <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big>
 
|-
 
!style="background:#efefef;"| <big>b</big>
 
| <big>b</big> || <big>c</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big>
 
|-
 
!style="background:#efefef;"| <big>c</big>
 
| <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big> || <big>a</big> || <big>b</big>
 
|-
 
!style="background:#efefef;"| <big>d</big>
 
| <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big>
 
|-
 
!style="background:#efefef;"| <big>f</big>
 
| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>  
 
|}
 

Текущая версия на 19:10, 4 сентября 2022

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TODO: НЕ ОЧЕНЬ ПОНИМАЮ, ЗАЧЕМ ВООБЩЕ ЭТО УТСВЕРЖДЕНИЕ ТУТ

Теорема:
Если $f$ — функция ограниченной вариации ($f \in \bigvee(a, b)$), то ее можно представить в виде разности монотонно неубывающих функций ($f = f_1 - f_2$).
Доказательство:
[math]\triangleright[/math]

Возьмем в качестве $f_1$ функцию $f_1(x) = \bigvee\limits_a^x (f)$, тогда по аддитивности она будет не убывать. Определим как $f_2$ функцию $f_2(x) = f_1(x) - f(x)$. Докажем, что она монотонно не убывает. $a < x_1 < x_2 < b$. Надо доказать, что $f_1(x_1) - f(x_1) \le f_1(x_2) - f(x_2)$, или что $f(x_2) - f(x_1) \le f_1(x_2) - f_1(x_1) = \bigvee\limits_{x_1}^{x_2} (f)$ (используем утверждение 1).

Но действительно $f(x_2) - f(x_1) \le
[math]\triangleleft[/math]

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