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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">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>









<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font>;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E53">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:55"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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="E1:55_1"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c7">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"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><a NAME="E2:55_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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">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="txt" href="mesfunc5.html#E2:55"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E3:55"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/><a NAME="E3:55_1"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67" title="EXTREAL2:th.67">EXTREAL2:67</a></i>;<br/><a NAME="E4:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal1.html#T27" title="EXTREAL1:th.27">EXTREAL1:27</a>, <a class="ref" href="extreal1.html#T28" title="EXTREAL1:th.28">EXTREAL1:28</a></i>;<br/><a NAME="E5:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/><a NAME="E6:55_1"/><b>then </b>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <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="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><a NAME="E7:55_1"/><b>then </b>
(  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x</font> <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="supinf_2.html#D3" title="SUPINF_2:def.3">SUPINF_2:def 3</a></i>;<br/><b>hence </b><a NAME="E8:55_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"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">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="extreal2.html#T59" title="EXTREAL2:th.59">EXTREAL2:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:55"/><b>then </b><i><font color="Green" title="E55">A5</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font> is <a href="seq_1.html#V1" title="SEQ_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E7:55"/><i><font color="Green" title="E56">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">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="c7">G</font></span>)</span>
 <b>and </b><br/><a NAME="E8:55"/><i><font color="Green" title="E57">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</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">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">G</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b3">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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="E1:55_2"/><i><font color="Green" title="E58">A8</font></i>: 
( <font color="Maroon" title="c8">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">G</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> )
 ;<br/><a NAME="E2:55_2"/><b>then </b>
<font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:55_2"/><b>then </b>
( <font color="Maroon" title="c9">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"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c10">y</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"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:55"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="mesfunc5.html#E7:55"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E1:55_2"><i><font color="Green" title="E58">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E4:55_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/><b>hence </b><a NAME="E5:55_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55_2"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="mesfunc5.html#E8:55"><i><font color="Green" title="E57">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:55"/>
<font color="Maroon" title="c5">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:55"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a>, <a class="txt" href="mesfunc5.html#E7:55"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="mesfunc5.html#E2:55"><i><font color="Green" title="E53">A2</font></i></a></i>;<br/>


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