<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>for <font color="Olive" title="b3">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b4">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">B</font>;<br/>
<a NAME="E1:30_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="ref" href="measure3.html#D4" target="_self" title="MEASURE3:def.4">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">B</font>
;<br/>

<b>thus </b><a NAME="E4:30_1_1_1"/>
( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">B</font>] )
 <b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="measure3.html#T22" target="_self" title="MEASURE3:th.22">Th22</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">comM</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">comM</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="measure3.html#E1:30_1_1"><i><font color="Green" title="E21">A1</font></i></a>);<br/></i>
<a NAME="E4:30_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c4">comM</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> holds   <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:30_1_1_2_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="ref" href="measure3.html#D4" target="_self" title="MEASURE3:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="measure3.html#E2:30_1_1_2_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E4:30_1_1_2_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="measure3.html#E2:30_1_1_2_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E5:30_1_1_2_1"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1_2_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_2_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_1_1_2"/>
<font color="Maroon" title="c4">comM</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:30_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c4">comM</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1_3"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">C</font> be    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>;<br/>

<a NAME="E2:30_1_1_3"/>
<font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_3"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="measure3.html#D4" target="_self" title="MEASURE3:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1_1_3"/>
<font color="Maroon" title="c4">comM</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="measure3.html#E1:30_1_1_3"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:30_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_4"/><i><font color="Green" title="E25">A9</font></i>: 
 ex <font color="Olive" title="b1">B1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_4_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
;<br/>

<b>thus </b><a NAME="E1:30_1_1_4_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a>, <a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1_4"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1_1_4"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_4"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="measure3.html#D3" target="_self" title="MEASURE3:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> ;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E8:30_1_1"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></i>;<br/>
<a NAME="E9:30_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">C</font>
 
;<br/>
<b>then </b><i><font color="Green" title="E26">A11</font></i>: <font color="Maroon" title="c4">comM</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="measure3.html#E5:30_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="measure3.html#E8:30_1_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>
;<br/>
<a NAME="E11:30_1_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> holds   <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">F</font> be   <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>;<br/>

<a NAME="E1:30_1_1_6"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:30_1_1_6"/><i><font color="Green" title="E28">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K3" title="MEASURE3:func.3">MeasPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure3.html#T18" target="_self" title="MEASURE3:th.18">Th18</a></i>;<br/>
<a NAME="E4:30_1_1_6"/><i><font color="Green" title="E29">A14</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:30_1_1_6_1"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K3" title="MEASURE3:func.3">MeasPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_6"><i><font color="Green" title="E28">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_6_1"/>
( <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="ref" href="measure3.html#D5" target="_self" title="MEASURE3:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:30_1_1_6"/><i><font color="Green" title="E30">A15</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="measure3.html#T19" target="_self" title="MEASURE3:th.19">Th19</a></i>;<br/>
<a NAME="E7:30_1_1_6"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:30_1_1_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_6_2"/>
<font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E9:30_1_1_6"/><i><font color="Green" title="E31">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure3.html#T20" target="_self" title="MEASURE3:th.20">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:30_1_1_6"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span>
 ;<br/>
<a NAME="E12:30_1_1_6"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_6_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_6_3"/><i><font color="Green" title="E34">A19</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Z</font> &amp; <font color="Maroon" title="c13">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E11:30_1_1_6"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:30_1_1_6_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:30_1_1_6_3"/><i><font color="Green" title="E35">A20</font></i>: 
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c13">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_6_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">n</font> = <font color="Maroon" title="c14">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6_3"><i><font color="Green" title="E35">A20</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">P</font> = <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font>;<br/>
<a NAME="E8:30_1_1_6_3"/><i><font color="Green" title="E36">A21</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>
<a NAME="E9:30_1_1_6_3"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font>
;<br/>

<b>thus </b><a NAME="E1:30_1_1_6_3_1"/>
( <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="measure3.html#E3:30_1_1_6_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="measure3.html#E6:30_1_1_6_3"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="measure3.html#E8:30_1_1_6_3"><i><font color="Green" title="E36">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:30_1_1_6_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:30_1_1_6"/><i><font color="Green" title="E34">A22</font></i>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">B</font>;<br/>
<i><font color="Green" title="E35">A23</font></i>:  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c11">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>
<br/>.= 
<font color="Maroon" title="c11">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">B</font></span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E12:30_1_1_6"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c13">C</font> = <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:30_1_1_6_4"/>
<font color="Maroon" title="c13">C</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_6_4_2"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_6_4_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 ;<br/>

<a NAME="E2:30_1_1_6_4_2_1"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> &amp; not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure3.html#E11:30_1_1_6"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:30_1_1_6_4_2_1"/><i><font color="Green" title="E38">A26</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font> &amp; <font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:30_1_1_6_4_2_1"/><i><font color="Green" title="E39">A27</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="measure3.html#E4:30_1_1_6_4_2_1"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6_4_2_1"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/>
<a NAME="E8:30_1_1_6_4_2_1"/>
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1_6_4_2_1_1"/><i><font color="Green" title="E40">A28</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font>
 ;<br/>

<a NAME="E2:30_1_1_6_4_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:30_1_1_6_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_6_4_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure3.html#E2:30_1_1_6_4_2_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="measure3.html#E1:30_1_1_6_4_2_1_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_1_1_6_4_2_1"/><b>then </b><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6_4_2_1"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="measure3.html#E6:30_1_1_6_4_2_1"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:30_1_1_6_4_2_1"/>
 ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_6_4_2_1_2"/><i><font color="Green" title="E41">A30</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
;<br/>

<b>thus </b><a NAME="E2:30_1_1_6_4_2_1_2"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="measure3.html#E9:30_1_1_6_4_2_1"><i><font color="Green" title="E40">A29</font></i></a>, <a class="txt" href="measure3.html#E1:30_1_1_6_4_2_1_2"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:30_1_1_6_4_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1_6_4_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_6_4_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_6_4_2_2"/><i><font color="Green" title="E37">A31</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font> &amp; <font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">H</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:30_1_1_6_4_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:30_1_1_6_4_2_2"/><i><font color="Green" title="E38">A32</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font> )
 <b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_6_4_2_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6_4_2_2"><i><font color="Green" title="E38">A32</font></i></a></i>;<br/>
<a NAME="E8:30_1_1_6_4_2_2"/><i><font color="Green" title="E39">A33</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> )
 
<b>by </b><i><a class="txt" href="measure3.html#E9:30_1_1_6"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="measure3.html#E3:30_1_1_6_4_2_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="measure3.html#E6:30_1_1_6_4_2_2"><i><font color="Green" title="E38">A32</font></i></a></i>;<br/>
<a NAME="E9:30_1_1_6_4_2_2"/>
<font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E6:30_1_1_6_4_2_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:30_1_1_6_4_2_2"/><b>then </b>
<font color="Maroon" title="c10">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:30_1_1_6_4_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E8:30_1_1_6_4_2_2"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1_6_4_2"/><b>then </b><i><font color="Green" title="E37">A34</font></i>: 
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_4_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:30_1_1_6_4_2"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_6_4_2_3"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_4_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1_6_4_2_3_1"/><i><font color="Green" title="E38">A35</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font>
 ;<br/>

<a NAME="E2:30_1_1_6_4_2_3_1"/><i><font color="Green" title="E39">A36</font></i>: 
 <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></i>;<br/>
<a NAME="E3:30_1_1_6_4_2_3_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:30_1_1_6_4_2_3_1"/><b>then </b><i><font color="Green" title="E40">A37</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_4_2_3_1"><i><font color="Green" title="E38">A35</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c14">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_4_2_3_1"><i><font color="Green" title="E38">A35</font></i></a>, <a class="txt" href="measure3.html#E2:30_1_1_6_4_2_3_1"><i><font color="Green" title="E39">A36</font></i></a></i>;<br/>
<a NAME="E6:30_1_1_6_4_2_3_1"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure3.html#E9:30_1_1_6"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="measure3.html#E4:30_1_1_6_4_2_3_1"><i><font color="Green" title="E40">A37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:30_1_1_6_4_2_3_1"/>
<font color="Maroon" title="c14">A</font> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1_6_4_2_3"/><b>then </b>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="measure2.html#T16" title="MEASURE2:th.16">MEASURE2:16</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1_1_6_4_2_3"/>
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30_1_1_6_4_2"/>
<font color="Maroon" title="c13">C</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_6_4_2"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="measure3.html#D3" target="_self" title="MEASURE3:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c14">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font><b> such that </b><br/><a NAME="E5:30_1_1_6_4"/><i><font color="Green" title="E36">A38</font></i>: 
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">C</font>
 <b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_4"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">C</font>
;<br/>

<b>thus </b><a NAME="E6:30_1_1_6_4"/>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">C</font>
 <b>by </b><i><a class="txt" href="measure3.html#E5:30_1_1_6_4"><i><font color="Green" title="E36">A38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:30_1_1_6"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1_6_5"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

<a NAME="E2:30_1_1_6_5"/><b>then </b>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>
<a NAME="E3:30_1_1_6_5"/><b>then </b><i><font color="Green" title="E35">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green" title="E36">A40</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">G</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E5:30_1_1_6_5"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_6_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_6_5_2_1"/>
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> or  not <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_6_5_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_6_5_2_1_1"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_6_5_2_1_1"/>
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_6_5_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_6_5_2_1_2"/><i><font color="Green" title="E37">A41</font></i>: 
not <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6_5"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="measure3.html#E1:30_1_1_6_5_2_1_2"><i><font color="Green" title="E37">A41</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:30_1_1_6_5_2_1_2"/>
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:30_1_1_6_5"/><b>then </b>
( <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> )
 
;<br/>
<a NAME="E7:30_1_1_6_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="measure3.html#E3:30_1_1_6_5"><i><font color="Green" title="E35">A39</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E8:30_1_1_6_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E9:30_1_1_6_5"/>
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">G</font> = <font color="Maroon" title="c8">G</font> as   <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>
<a NAME="E16:30_1_1_6"/><i><font color="Green" title="E35">A42</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E17:30_1_1_6"/><i><font color="Green" title="E36">A43</font></i>: 
<font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="measure1.html#T54" title="MEASURE1:th.54">MEASURE1:54</a></i>;<br/>
<a NAME="E18:30_1_1_6"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E19:30_1_1_6"/><b>then </b><i><font color="Green" title="E37">A44</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#T54" title="MEASURE1:th.54">MEASURE1:54</a></i>;<br/>
<a NAME="E20:30_1_1_6"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_6"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_6_6"/><i><font color="Green" title="E38">A45</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:30_1_1_6_6_1"/><i><font color="Green" title="E39">A46</font></i>: 
( <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6"><i><font color="Green" title="E29">A14</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font><b> such that </b><br/><a NAME="E3:30_1_1_6_6_1"/><i><font color="Green" title="E40">A47</font></i>: 
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 ;<br/>
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">C</font>
<b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_6_1"><i><font color="Green" title="E39">A46</font></i></a>, <a class="txt" href="measure3.html#E3:30_1_1_6_6_1"><i><font color="Green" title="E40">A47</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E5:30_1_1_6_6_1"/>
<font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E5:30_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30_1_1_6_6"/><i><font color="Green" title="E39">A48</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c13">n</font> = 
<font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="txt" href="measure3.html#E1:30_1_1_6_6"><i><font color="Green" title="E38">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<b>hence </b><a NAME="E2:30_1_1_6_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1_6_6"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<a NAME="E4:30_1_1_6_6"/><b>then </b><i><font color="Green" title="E40">A49</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E17:30_1_1_6"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<a NAME="E5:30_1_1_6_6"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="measure3.html#E2:30_1_1_6_6"><i><font color="Green" title="E39">A48</font></i></a></i>;<br/>
<a NAME="E6:30_1_1_6_6"/><b>then </b>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E19:30_1_1_6"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="supinf_2.html#T62" title="SUPINF_2:th.62">SUPINF_2:62</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1_1_6_6"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1_6_6"><i><font color="Green" title="E40">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:30_1_1_6"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">comM</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">comM</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E5:30_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="measure3.html#E11:30_1_1_6"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="measure3.html#E13:30_1_1_6"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="measure3.html#E16:30_1_1_6"><i><font color="Green" title="E35">A42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">comM</font> = <font color="Maroon" title="c4">comM</font> as    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="measure3.html#E4:30_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="measure3.html#E10:30_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">comM</font>
;<br/>

<b>thus </b><a NAME="E13:30_1_1"/>
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c7">comM</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">B</font>
 <b>by </b><i><a class="txt" href="measure3.html#E5:30_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>


</div>
