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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:55"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:55"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:55"><i><font color="Green">E52</font></i></a>, <a class="ref" href="mesfunc2.html#D5">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">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:55_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:55"><i><font color="Green">E53</font></i></a>, <a class="txt" href="mesfunc5.html#E3:55"><i><font color="Green">E54</font></i></a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E3:55_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/><a NAME="E4:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal1.html#T27">EXTREAL1:27</a>, <a class="ref" href="extreal1.html#T28">EXTREAL1:28</a></i>;<br/><a NAME="E5:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/><a NAME="E6:55_1"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><a NAME="E7:55_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><b>hence </b><a NAME="E8:55_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T59">EXTREAL2:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:55"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:55"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:55"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55"><i><font color="Green">E52</font></i></a>, <a class="ref" href="mesfunc2.html#D5">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">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:55_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:55_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:55"><i><font color="Green">E53</font></i></a>, <a class="txt" href="mesfunc5.html#E7:55"><i><font color="Green">E56</font></i></a>, <a class="txt" href="mesfunc5.html#E1:55_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:55_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/><b>hence </b><a NAME="E5:55_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:55_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="mesfunc5.html#E8:55"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:55"/>
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:55"><i><font color="Green">E55</font></i></a>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a>, <a class="txt" href="mesfunc5.html#E7:55"><i><font color="Green">E56</font></i></a>, <a class="txt" href="mesfunc5.html#E2:55"><i><font color="Green">E53</font></i></a></i>;<br/>


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