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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="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>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc2.html#R1" target="_self">is_finite</a> 
 ;<br/>

<a NAME="E2:11"/>
for <font color="Olive">b<sub>1</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><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>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:11_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</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>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:11_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>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</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="mesfunc2.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mesfunc2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:11_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> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</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="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>
<a NAME="E5:11_1"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</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>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E7:11_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</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="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<a NAME="E8:11_1"/><b>then </b><i><font color="Green">E13</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="extreal1.html#K2">*</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> &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="extreal1.html#K2">*</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal1.html#T38">EXTREAL1:38</a></i>;<br/>
<a NAME="E9:11_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="extreal1.html#K2">*</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc2.html#E1:11_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E10:11_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>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</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="txt" href="mesfunc2.html#E8:11_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:11_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>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</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>
<b>hence </b><a NAME="E3:11"/>
<font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc2.html#R1" target="_self">is_finite</a> 
 <b>by </b><i><a class="ref" href="mesfunc2.html#D1" target="_self">Def1</a></i>;<br/>


</div>
