<?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>, <font color="Maroon">c<sub>6</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="E1:114"/><i><font color="Green">E108</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>5</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>then 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="E3:114"/><i><font color="Green">E109</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>
 <b>and </b><br/><a NAME="E4:114"/><i><font color="Green">E110</font></i>: 
<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>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<a NAME="E5:114"/><i><font color="Green">E111</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>5</sub></font> <a href="measure1.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>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:114"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</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>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:114"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</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"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:114"/><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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E5:114"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:114"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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="114_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_1"/><b>then </b>
<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>5</sub></font> <a href="measure1.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>9</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>9</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="E3:114_1"/>
<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>5</sub></font> <a href="measure1.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>9</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>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E8:114"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_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="E10:114"/><b>then </b><i><font color="Green">E115</font></i>: 
<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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.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#E8:114"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E11:114"/>
<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> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E6:114"><i><font color="Green">E112</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green">E109</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114"><i><font color="Green">E110</font></i></a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E12:114"/><b>then </b><i><font color="Green">E116</font></i>: 
<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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E10:114"><i><font color="Green">E115</font></i></a>, <a class="txt" href="mesfunc5.html#E7:114"><i><font color="Green">E113</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E13:114"/><i><font color="Green">E117</font></i>: 
<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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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>10</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E15:114"/><i><font color="Green">E118</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>9</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>9</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> ) ) &amp; ( 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="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( 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>9</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>9</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> ) &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/>( <font color="Maroon">c<sub>9</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>9</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> ) ) &amp; ( 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="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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp; <font color="Maroon">c<sub>10</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>10</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green">E108</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</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="E17:114"/><i><font color="Green">E119</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="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>11</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>12</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E19:114"/><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>12</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="E20:114"/><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 <br/>( <font color="Maroon">c<sub>11</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>11</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</sub></font> ) &amp; ( 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="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> ) &amp; ( 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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>11</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> ) &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> <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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/>( <font color="Maroon">c<sub>11</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>11</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</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="114_2"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green">E119</font></i></a></i>;<br/><a NAME="E2:114_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/><b>hence </b><a NAME="E3:114_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_2_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#T40" target="_self">Th40</a></i>;<br/><b>thus </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
 <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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.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#E17:114"><i><font color="Green">E119</font></i></a></i>
<br/>.= 
<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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<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="E1:114_2_2"/>
<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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_2_2"/><b>then </b><i><font color="Green">E124</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>11</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#E1:114_2"><i><font color="Green">E122</font></i></a></i>;<br/><a NAME="E3:114_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green">E119</font></i></a></i>;<br/><b>hence </b><a NAME="E4:114_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>9</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#E2:114_2_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:114_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/><a NAME="E2:114_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> 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="E3:114_2_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:114"><i><font color="Green">E119</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:114_2_4"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</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="E1:114_2_4_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>15</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>15</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E3:114_2_4_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>15</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="E4:114_2_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_2_4_1"><i><font color="Green">E125</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_2_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_2_4_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_2_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:114_2_4"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>13</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="E1:114_2_5"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>13</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:114_2_5"/><b>then </b>
<font color="Maroon">c<sub>13</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E3:114_2_5"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>13</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/><div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mesfunc5.html#K3" target="_self">#</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> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_2_5"><i><font color="Green">E124</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</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="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div><a NAME="E5:114_2_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>13</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="E6:114_2_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>13</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_2_5"><i><font color="Green">E125</font></i></a></i>;<br/><b>thus </b> <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
 <a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_2_5"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_2_5"><i><font color="Green">E125</font></i></a></i>
<br/>.= 
<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>5</sub></font> <a href="measure1.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>13</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_2_5"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E21:114"/>
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/><font color="Maroon">c<sub>12</sub></font> <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>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:114_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green">E121</font></i></a></i>;<br/>
<a NAME="E3:114_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:114"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E4:114_3"/><i><font color="Green">E125</font></i>: 
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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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="114_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_3_1"/>
<font color="Maroon">c<sub>15</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"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_3_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <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#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_3_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</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>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green">E121</font></i></a>, <a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_3"><i><font color="Green">E122</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:114_3"/><b>then </b><i><font color="Green">E126</font></i>: 
 <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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E6:114_3"/>
 <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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</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"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E7:114_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:114_3"><i><font color="Green">E123</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:114_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E19:114"><i><font color="Green">E120</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_3"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114_3"><i><font color="Green">E126</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:114"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E23:114"/><b>then </b><i><font color="Green">E122</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"><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>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a>, <a class="txt" href="mesfunc5.html#E20:114"><i><font color="Green">E121</font></i></a>, <a class="txt" href="mesfunc5.html#E19:114"><i><font color="Green">E120</font></i></a>, <a class="txt" href="mesfunc5.html#E12:114"><i><font color="Green">E116</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114"><i><font color="Green">E111</font></i></a>, <a class="txt" href="mesfunc5.html#E13:114"><i><font color="Green">E117</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E24:114"/>
( <font color="Maroon">c<sub>7</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>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E25:114"/><b>then </b><i><font color="Green">E123</font></i>: 
( <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> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</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> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:114"><i><font color="Green">E110</font></i></a>, <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E26:114"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <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"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E27:114"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</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="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E28:114"/><b>then </b><i><font color="Green">E125</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>7</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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E29:114"/><b>then </b><i><font color="Green">E126</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>7</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> &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="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</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>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E124</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E30:114"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> holds <br/><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>7</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> <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="114_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>15</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_4"/><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>15</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>15</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:114_4"/>
<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>7</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>15</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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_4"><i><font color="Green">E127</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:114"/><b>then </b>
<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>7</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>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E32:114"/><b>then </b><i><font color="Green">E127</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>7</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#E28:114"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E25:114"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E33:114"/>
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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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> <font color="Maroon">c<sub>6</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="114_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_5"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>15</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_5"/><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>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</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>15</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:114_5"/>
<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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</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> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_5"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:114"/><b>then </b>
<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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E29:114"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E35:114"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <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>7</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:114"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E25:114"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E36:114"/><i><font color="Green">E129</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> &amp; <font color="Maroon">c<sub>4</sub></font> <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="txt" href="mesfunc5.html#E1:114"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</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>9</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>15</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="E38:114"/><i><font color="Green">E130</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>15</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>3</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>4</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>9</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>6</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</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="E40:114"/><i><font color="Green">E131</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>16</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>4</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>
<a NAME="E41:114"/><i><font color="Green">E132</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>15</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; <font color="Maroon">c<sub>16</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>15</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</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>6</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:114_6"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>
<a NAME="E2:114_6"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:114_6"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>, <a class="txt" href="mesfunc5.html#E1:114_6"><i><font color="Green">E133</font></i></a></i>;<br/>

<a NAME="E4:114_6"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_6"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E5:114_6"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114"><i><font color="Green">E109</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:114"/><i><font color="Green">E133</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>15</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> &amp; <font color="Maroon">c<sub>16</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="114_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:114_7"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>
<a NAME="E2:114_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</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/>
<b>hence </b><a NAME="E3:114_7"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:114"/><i><font color="Green">E134</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>15</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>15</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="114_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_8"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</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="E2:114_8"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>19</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="E3:114_8"/><b>then </b>
<font color="Maroon">c<sub>19</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="E4:114_8"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>19</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="E5:114_8"/>
(  <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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E6:114_8"/><b>then </b><i><font color="Green">E138</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</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#E41:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E7:114_8"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>19</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</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>19</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E8:114_8"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_8"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E6:114_8"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E9:114_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_8"><i><font color="Green">E135</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_8"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:114"/><i><font color="Green">E135</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>6</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</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>16</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="114_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_9"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</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="E2:114_9"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>19</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>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:114_9"/><b>then </b>
<font color="Maroon">c<sub>19</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>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:114_9"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>19</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="E5:114_9"/>
(  <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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E6:114_9"/><b>then </b><i><font color="Green">E139</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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>6</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</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>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E7:114_9"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</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>19</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</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>19</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E8:114_9"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_9"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E6:114_9"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E9:114_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_9"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_9"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:114"/><i><font color="Green">E136</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>15</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>15</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="114_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:114_10"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>17</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="E2:114_10"/><b>then </b>
<font color="Maroon">c<sub>17</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="E3:114_10"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>17</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E139</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</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#E41:114"><i><font color="Green">E132</font></i></a></i>
;<br/><a NAME="E2:114_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:114_10_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</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>17</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a></i>;<br/><a NAME="E4:114_10_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_10"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_10_1"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E5:114_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</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="E5:114_10"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_10"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>17</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_10"><i><font color="Green">E138</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_10"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>15</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>17</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:114_10"><i><font color="Green">E139</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_10"><i><font color="Green">E137</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:114"/><i><font color="Green">E137</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>6</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>16</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>16</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>6</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="114_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:114_11"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>17</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>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:114_11"/><b>then </b>
<font color="Maroon">c<sub>17</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>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:114_11"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>17</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E140</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</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>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</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>17</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_11"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_11_1"><i><font color="Green">E140</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>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</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="E5:114_11"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E6:114_11"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>17</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_11"><i><font color="Green">E139</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:114_11"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>16</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>17</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>6</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:114_11"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_11"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>5</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>15</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>17</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E48:114"/><i><font color="Green">E138</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>17</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>5</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>
<b>deffunc </b><font color="Maroon">H<sub>6</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>16</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>18</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E50:114"/><i><font color="Green">E139</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>18</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>6</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="E51:114"/>
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/>( <font color="Maroon">c<sub>17</sub></font> <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>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <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>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:114_12"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E2:114_12"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</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"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</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>6</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a></i>;<br/>
<a NAME="E3:114_12"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E4:114_12"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E48:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green">E139</font></i></a></i>;<br/>
<a NAME="E5:114_12"/><i><font color="Green">E144</font></i>: 
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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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="114_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_12_1"/>
<font color="Maroon">c<sub>21</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"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_12_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_12_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_12"><i><font color="Green">E140</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:114_12"/><b>then </b><i><font color="Green">E145</font></i>: 
 <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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_12"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E7:114_12"/>
 <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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_12"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E5:114_12"><i><font color="Green">E144</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E8:114_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:114_12"/>
<font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E48:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_12"><i><font color="Green">E143</font></i></a>, <a class="txt" href="mesfunc5.html#E6:114_12"><i><font color="Green">E145</font></i></a></i>;<br/>

<a NAME="E10:114_12"/><i><font color="Green">E146</font></i>: 
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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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="114_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:114_12_2"/>
<font color="Maroon">c<sub>21</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"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:114_12_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E3:114_12_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:114_12_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E44:114"><i><font color="Green">E135</font></i></a>, <a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114_12"><i><font color="Green">E140</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:114_12"/><b>then </b><i><font color="Green">E147</font></i>: 
 <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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_12"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E12:114_12"/>
 <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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</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"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E3:114_12"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E10:114_12"><i><font color="Green">E146</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E13:114_12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:114_12"><i><font color="Green">E141</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:114_12"/>
<font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green">E139</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_12"><i><font color="Green">E143</font></i></a>, <a class="txt" href="mesfunc5.html#E11:114_12"><i><font color="Green">E147</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E52:114"/><b>then </b><i><font color="Green">E140</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/><font color="Maroon">c<sub>17</sub></font> <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>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( 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/><font color="Maroon">c<sub>18</sub></font> <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>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
;<br/>
<a NAME="E53:114"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E54:114"/><b>then </b><i><font color="Green">E141</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"><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>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <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>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a>, <a class="txt" href="mesfunc5.html#E26:114"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E32:114"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E35:114"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E36:114"><i><font color="Green">E129</font></i></a>, <a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a>, <a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E133</font></i></a>, <a class="txt" href="mesfunc5.html#E43:114"><i><font color="Green">E134</font></i></a>, <a class="txt" href="mesfunc5.html#E45:114"><i><font color="Green">E136</font></i></a>, <a class="txt" href="mesfunc5.html#E48:114"><i><font color="Green">E138</font></i></a>, <a class="txt" href="mesfunc5.html#E44:114"><i><font color="Green">E135</font></i></a>, <a class="txt" href="mesfunc5.html#E46:114"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green">E139</font></i></a></i>;<br/>
<a NAME="E55:114"/><i><font color="Green">E142</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>12</sub></font> <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>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:114_13"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:114"><i><font color="Green">E118</font></i></a></i>;<br/>
<a NAME="E2:114_13"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E38:114"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E40:114"><i><font color="Green">E131</font></i></a></i>;<br/>
<a NAME="E3:114_13"/><b>then </b><i><font color="Green">E144</font></i>: 
 <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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:114_13"><i><font color="Green">E143</font></i></a>, <a class="txt" href="mesfunc5.html#E1:114"><i><font color="Green">E108</font></i></a>, <a class="ref" href="mesfunc5.html#T73" target="_self">Th73</a></i>;<br/>
<i><font color="Green">E145</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> = 
 <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>11</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E19:114"><i><font color="Green">E120</font></i></a></i>
<br/>.= 
 <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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.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#E17:114"><i><font color="Green">E119</font></i></a></i>
;<br/>
<a NAME="E5:114_13"/>
<font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E48:114"><i><font color="Green">E138</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:114_13"/>
<font color="Maroon">c<sub>12</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_13"><i><font color="Green">E144</font></i></a>, <a class="txt" href="mesfunc5.html#E4:114_13"><i><font color="Green">E145</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green">E139</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:114"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <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>17</sub></font> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <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>18</sub></font> implies  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</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="114_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:114_14_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:114_14_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</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>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a>, <a class="txt" href="mesfunc5.html#E48:114"><i><font color="Green">E138</font></i></a></i>;<br/><a NAME="E4:114_14_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E133</font></i></a></i>;<br/><b>hence </b><a NAME="E5:114_14_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:114_14_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="mesfunc5.html#T74" target="_self">Th74</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:114_14"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E4:114_14"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>20</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>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E41:114"><i><font color="Green">E132</font></i></a>, <a class="txt" href="mesfunc5.html#E50:114"><i><font color="Green">E139</font></i></a></i>;<br/>
<a NAME="E5:114_14"/>
<font color="Maroon">c<sub>16</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:114"><i><font color="Green">E133</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:114_14"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:114_14"><i><font color="Green">E143</font></i></a>, <a class="ref" href="mesfunc5.html#T74" target="_self">Th74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E57:114"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E58:114"/>
 <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<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></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <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#E23:114"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E54:114"><i><font color="Green">E141</font></i></a>, <a class="ref" href="mesfunc5.html#T67" target="_self">Th67</a>, <a class="txt" href="mesfunc5.html#E52:114"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E55:114"><i><font color="Green">E142</font></i></a></i>;<br/>


</div>
