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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="E1:143"/><i><font color="Green">E128</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/>(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <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> &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> )
 ;<br/>

<b>then 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="E3:143"/><i><font color="Green">E129</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<a NAME="E4:143"/><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E5:143"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:143"/><b>then </b><i><font color="Green">E131</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:143"/><b>then </b><i><font color="Green">E132</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</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; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</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>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</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="E9:143"/><i><font color="Green">E133</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>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>7</sub></font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self">Def17</a>, <a class="txt" href="mesfunc5.html#E1:143"><i><font color="Green">E128</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E11:143"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>8</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>8</sub></font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" target="_self">Def17</a>, <a class="txt" href="mesfunc5.html#E1:143"><i><font color="Green">E128</font></i></a></i>;<br/>
<a NAME="E12:143"/>
( <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> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</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#E1:143"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E9:143"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E11:143"><i><font color="Green">E134</font></i></a>, <a class="ref" href="mesfunc5.html#T106" target="_self">Th106</a></i>;<br/>
<a NAME="E13:143"/><b>then </b><i><font color="Green">E135</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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#T103" target="_self">Th103</a></i>;<br/>
<a NAME="E14:143"/>
( <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> &amp; <font color="Maroon">c<sub>5</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#E9:143"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E11:143"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E6:143"><i><font color="Green">E131</font></i></a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E15:143"/><b>then </b><i><font color="Green">E136</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="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</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#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E6:143"><i><font color="Green">E131</font></i></a>, <a class="ref" href="mesfunc2.html#T29">MESFUNC2:29</a></i>;<br/>
<a NAME="E16:143"/>
( <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> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</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="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E17:143"/><b>then </b><i><font color="Green">E137</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E18:143"/><b>then </b><i><font color="Green">E138</font></i>: 
<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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</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#E13:143"><i><font color="Green">E135</font></i></a>, <a class="ref" href="mesfunc5.html#T112" target="_self">Th112</a></i>;<br/>
<a NAME="E19:143"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</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="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E20:143"/><b>then </b><i><font color="Green">E139</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">(<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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <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">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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#E15:143"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="mesfunc5.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E21:143"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E22:143"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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/><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>, <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E24:143"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>, <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</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="E1:143_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>11</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>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><a NAME="E2:143_1"/><b>then </b>
( <font color="Maroon">c<sub>11</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> &amp; <font color="Maroon">c<sub>11</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>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E3:143_1"/><b>then </b><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/><a NAME="E4:143_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E17:143"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E21:143"><i><font color="Green">E140</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:143_1"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_1_1"/><i><font color="Green">E145</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:143_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1_1"><i><font color="Green">E145</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1_1"><b>suppose </b></a><a NAME="E1:143_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:143_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:143_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E4:143_1_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:143_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:143_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143_1_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="143_1_1_1_2"><b>suppose </b></a><a NAME="E1:143_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:143_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:143_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:143_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E4:143_1_1_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:143_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:143_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E4:143_1_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:143_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<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>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</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>11</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T61">EXTREAL2:61</a></i>;<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="Maroon">c<sub>11</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> = 
<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>11</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</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>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green">E132</font></i></a></i>
<br/>.= 
<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>11</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E7:143"><i><font color="Green">E132</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E20:143"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E20:143"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:143_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
;<br/><b>hence </b><a NAME="E8:143_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>4</sub></font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:143_1"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a>, <a class="txt" href="mesfunc5.html#E6:143_1"><i><font color="Green">E145</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:143"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E21:143"><i><font color="Green">E140</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="txt" href="mesfunc5.html#E17:143"><i><font color="Green">E137</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:143"/>
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K13">+</a> <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>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E18:143"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E26:143"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E3:143"><i><font color="Green">E129</font></i></a>, <a class="ref" href="mesfunc5.html#T108" target="_self">Th108</a></i>;<br/>


</div>
