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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>, <font color="Maroon">c<sub>5</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>assume </b><a NAME="E1:137"/><i><font color="Green">E122</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">b<sub>1</sub></font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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> <font color="Maroon">c<sub>4</sub></font> holds <br/><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="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:137"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:137"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self">Def17</a>, <a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a></i>;<br/>
<a NAME="E6:137"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:137"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a>, <a class="ref" href="mesfunc2.html#T29">MESFUNC2:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:137_1"/>
<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>5</sub></font>
 ;<br/><a NAME="E2:137_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>8</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a></i>;<br/><b>hence </b><a NAME="E3:137_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51">EXTREAL2:51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:137"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:137_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span>
 ;<br/><a NAME="E2:137_2"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <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>8</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:137_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51">EXTREAL2:51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:137"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<a NAME="E11:137"/><i><font color="Green">E128</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="137_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:137_3"/><i><font color="Green">E130</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> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span>
 ;<br/><a NAME="E2:137_3"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <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>8</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/><a NAME="E3:137_3"/><b>then </b><i><font color="Green">E131</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="extreal2.html#T51">EXTREAL2:51</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,0
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:137_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a>, <a class="txt" href="mesfunc5.html#E3:137_3"><i><font color="Green">E131</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E13:137"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K1">max+</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:137_4"/><i><font color="Green">E131</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>5</sub></font>
 ;<br/><a NAME="E2:137_4"/><i><font color="Green">E132</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:137"><i><font color="Green">E126</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,0
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:137_4"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a>, <a class="txt" href="mesfunc5.html#E2:137_4"><i><font color="Green">E132</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E15:137"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a></i>;<br/>
<div><i><font color="Green">E132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="137_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:137_5"/><i><font color="Green">E133</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> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span>
 ;<br/><a NAME="E2:137_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E12:137"><i><font color="Green">E129</font></i></a></i>;<br/><b>hence </b><a NAME="E3:137_5"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:137_5"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc2.html#T21">MESFUNC2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:137"/>
 <a href="mesfunc2.html#K2">max-</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E18:137"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T93" target="_self">Th93</a>, <a class="txt" href="mesfunc5.html#E16:137"><i><font color="Green">E132</font></i></a>, <a class="txt" href="mesfunc5.html#E11:137"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a></i>;<br/>
<a NAME="E19:137"/>
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"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> holds <br/><span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:137_6"/><i><font color="Green">E134</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> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span>
 ;<br/>

<a NAME="E2:137_6"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <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>8</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:137_6"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E1:137_6"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E3:137"><i><font color="Green">E123</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:137"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T91" target="_self">Th91</a>, <a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E3:137"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E5:137"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E8:137"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E10:137"><i><font color="Green">E127</font></i></a></i>;<br/>
<a NAME="E21:137"/>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</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="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E22:137"/><b>then </b>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></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="txt" href="mesfunc5.html#E13:137"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E20:137"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="txt" href="mesfunc5.html#E15:137"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E23:137"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E18:137"><i><font color="Green">E133</font></i></a>, <a class="ref" href="mesfunc5.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E24:137"/>
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T106" target="_self">Th106</a>, <a class="txt" href="mesfunc5.html#E1:137"><i><font color="Green">E122</font></i></a></i>;<br/>

<a NAME="E25:137"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T94" target="_self">Th94</a>, <a class="txt" href="mesfunc5.html#E5:137"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E8:137"><i><font color="Green">E126</font></i></a></i>;<br/>
<b>hence </b><a NAME="E26:137"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T94" target="_self">Th94</a>, <a class="txt" href="mesfunc5.html#E6:137"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E10:137"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E20:137"><i><font color="Green">E134</font></i></a></i>;<br/>


</div>
