<?xml version="1.0"?>
<div class="add">

<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="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:115"/><i><font color="Green">E109</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; <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<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="E3:115"/><i><font color="Green">E110</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#E1:115"><i><font color="Green">E109</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E4:115"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:115"/><b>then </b><i><font color="Green">E112</font></i>: 
<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> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:115"><i><font color="Green">E110</font></i></a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E6:115"/><i><font color="Green">E113</font></i>: 
<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"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</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> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:115"><i><font color="Green">E111</font></i></a>, <a class="txt" href="mesfunc5.html#E3:115"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:115"/><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="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</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>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:115"/><b>then </b><i><font color="Green">E114</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="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:115"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:115"/>
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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </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="E1:115_1"/><i><font color="Green">E115</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"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:115_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</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="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:115_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</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#E8:115"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E1:115_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:115"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:115"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E11:115"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</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> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:115"><i><font color="Green">E113</font></i></a>, <a class="txt" href="mesfunc5.html#E5:115"><i><font color="Green">E112</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">c<sub>9</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E13:115"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</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>and </b><br/><a NAME="E14:115"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E15:115"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <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>3</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">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E16:115"/><i><font color="Green">E119</font></i>: 
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/>( <font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">b<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <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>1</sub></font> )
 <b>and </b><br/><a NAME="E17:115"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:115"><i><font color="Green">E109</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="seqfunc.html#M1">Functional_Sequence</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E19:115"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1">SEQFUNC:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  =  <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E21:115"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>11</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E22:115"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="measure1.html#K3">/\</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"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:115"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E23:115"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:115"><i><font color="Green">E109</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E24:115"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:115_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:115"><i><font color="Green">E116</font></i></a>, <a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:115_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</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="ref" href="mesfunc5.html#T40" target="_self">Th40</a></i>;<br/>

<a NAME="E3:115_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E4:115_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E5:115_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:115"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E3:115"><i><font color="Green">E110</font></i></a>, <a class="txt" href="mesfunc5.html#E13:115"><i><font color="Green">E116</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:115"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:115_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E14:115"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E2:115_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <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="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E3:115_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:115"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <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>3</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:115_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>14</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="E2:115_4"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>14</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:115_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</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> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:115_4"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>14</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:115_4"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E6:115_4"/><b>then </b><i><font color="Green">E130</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:115"><i><font color="Green">E124</font></i></a></i>;<br/>
<a NAME="E7:115_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E8:115_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:115_4"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E6:115_4"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E9:115_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:115_4"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E4:115_4"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E15:115"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:115"/><i><font color="Green">E127</font></i>: 
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>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">b<sub>1</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:115_5"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>12</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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:115_5"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:115_5"/><b>then </b>
<font color="Maroon">c<sub>12</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:115_5"/><b>then </b><i><font color="Green">E129</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E16:115"><i><font color="Green">E119</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E130</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>
<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="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E24:115"><i><font color="Green">E124</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:115_5"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E1:115_5_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:115_5"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E7:115_5"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:115_5"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E1:115_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:115"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>11</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:115_6"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:115"><i><font color="Green">E116</font></i></a>, <a class="txt" href="mesfunc5.html#E14:115"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E2:115_6"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T79" target="_self">Th79</a>, <a class="txt" href="mesfunc5.html#E1:115"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E3:115_6"/><b>then </b>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:115"><i><font color="Green">E120</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:115_6"/>
<font color="Maroon">c<sub>11</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E21:115"><i><font color="Green">E121</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:115"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> 0 )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T66" target="_self">Th66</a></i>;<br/>
<b>hence </b><a NAME="E30:115"/>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:115"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E11:115"><i><font color="Green">E115</font></i></a>, <a class="txt" href="mesfunc5.html#E23:115"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a>, <a class="txt" href="mesfunc5.html#E24:115"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E25:115"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E26:115"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E27:115"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E21:115"><i><font color="Green">E121</font></i></a></i>;<br/>


</div>
