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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>





<a NAME="E1:32"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c4">x</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> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 <b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a></i>;<br/><a NAME="E3:32_1_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:32_1_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</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="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32"/>
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">X</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">finite</a>
 <b>by </b><i><a class="ref" href="mesfunc2.html#D1" target="_self" title="MESFUNC2:def.1">Def1</a></i>;<br/>


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