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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:119"/><i><font color="Green">E113</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; 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> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</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>7</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E3:119"/><i><font color="Green">E114</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>6</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>6</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>6</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>6</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>6</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>6</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>6</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>7</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp; <font color="Maroon">c<sub>7</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>7</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:119_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:119_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/><b>hence </b><a NAME="E4:119_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T74" target="_self">Th74</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:119"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>7</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/>
<a NAME="E6:119"/><i><font color="Green">E116</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>7</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>7</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="119_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:119_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:119_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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; <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E3:119_2"/><i><font color="Green">E119</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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="119_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:119_2_1"/>
<font color="Maroon">c<sub>10</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:119_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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="E3:119_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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="E4:119_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E2:119_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E1:119_2"><i><font color="Green">E117</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:119_2"/><b>then </b>
 <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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:119_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="mesfunc5.html#T75" target="_self">Th75</a></i>;<br/>
<a NAME="E5:119_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mesfunc1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a>, <a class="txt" href="mesfunc5.html#E2:119_2"><i><font color="Green">E118</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:119_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:119_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:119_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="mesfunc5.html#T76" target="_self">Th76</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:119"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a></i>;<br/>
<a NAME="E9:119"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>8</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a>, <a class="ref" href="mesfunc1.html#T41">MESFUNC1:41</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E10:119"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</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>  = <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>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> <b> such that </b><br/><a NAME="E12:119"/><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>9</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>9</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>10</sub></font> being   <a href="mesfunc5.html#NM1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E14:119"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</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="E15:119"/><i><font color="Green">E122</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:119_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E2:119_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:119_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</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#E1:119_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="mesfunc5.html#T45" target="_self">Th45</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:119_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a>, <a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E16:119"/><i><font color="Green">E123</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>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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:119_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E2:119_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:119_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:119"/><i><font color="Green">E124</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:119_5"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<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="E2:119_5"/><i><font color="Green">E126</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:119_5"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E4:119_5"/><b>then </b><i><font color="Green">E127</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:119"><i><font color="Green">E122</font></i></a></i>;<br/>
<a NAME="E5:119_5"/>
( <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>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> &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>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</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#E12:119"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E6:119_5"/><b>then </b><i><font color="Green">E128</font></i>: 
( <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>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &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>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:119_5"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E4:119_5"><i><font color="Green">E127</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E7:119_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_5"><i><font color="Green">E125</font></i></a>, <a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E2:119_5"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E9:119"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:119_5"/>
<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>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</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="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:119_5"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:119"/><i><font color="Green">E125</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</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="119_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:119_6"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<div><i><font color="Green">E127</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><i><font color="Green">E128</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> = 
 <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="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E15:119"><i><font color="Green">E122</font></i></a></i>
;<br/><b>thus </b><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>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<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>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_6"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E1:119_6_1"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:119_6_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:119_6_2"/>
<font color="Maroon">c<sub>6</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#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/><a NAME="E4:119_6_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is <a href="mesfunc5.html#V3" target="_self">without-infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E5:119_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</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>6</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>11</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>;<br/><b>hence </b><a NAME="E6:119_6_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:119_6_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="mesfunc5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:119_6"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</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/>
<div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:119_6_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:119_6_3"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:119_6_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</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>6</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>11</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" target="_self">Def13</a></i>;<br/><b>hence </b><a NAME="E4:119_6_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</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#E1:119_6"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E1:119_6_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E2:119_6_3"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a>, <a class="txt" href="mesfunc5.html#E9:119"><i><font color="Green">E118</font></i></a></i>;<br/></div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_6"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E8:119"><i><font color="Green">E117</font></i></a>, <a class="txt" href="mesfunc5.html#E9:119"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_6"><i><font color="Green">E126</font></i></a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E7:119_6"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="mesfunc5.html#K3" target="_self">#</a> <font color="Maroon">c<sub>11</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <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>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T69" target="_self">Th69</a>, <a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="txt" href="mesfunc5.html#E2:119_6"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E4:119_6"><i><font color="Green">E128</font></i></a>, <a class="txt" href="mesfunc5.html#E5:119_6"><i><font color="Green">E129</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:119"/>
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> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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="119_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:119_7"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>;<br/>
<a NAME="E2:119_7"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:119"><i><font color="Green">E120</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</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>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E14:119"><i><font color="Green">E121</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.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>6</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:119_7"><i><font color="Green">E126</font></i></a>, <a class="txt" href="mesfunc5.html#E2:119_7"><i><font color="Green">E127</font></i></a>, <a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="ref" href="mesfunc5.html#T72" target="_self">Th72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:119"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> is <a href="mesfunc5.html#V10" target="_self">convergent</a> &amp;  <a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T69" target="_self">Th69</a>, <a class="txt" href="mesfunc5.html#E1:119"><i><font color="Green">E113</font></i></a>, <a class="txt" href="mesfunc5.html#E5:119"><i><font color="Green">E115</font></i></a>, <a class="txt" href="mesfunc5.html#E6:119"><i><font color="Green">E116</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:119"/>
 <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>5</sub></font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="extreal1.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>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D15" target="_self">Def15</a>, <a class="txt" href="mesfunc5.html#E3:119"><i><font color="Green">E114</font></i></a>, <a class="txt" href="mesfunc5.html#E9:119"><i><font color="Green">E118</font></i></a>, <a class="txt" href="mesfunc5.html#E10:119"><i><font color="Green">E119</font></i></a>, <a class="txt" href="mesfunc5.html#E14:119"><i><font color="Green">E121</font></i></a>, <a class="txt" href="mesfunc5.html#E15:119"><i><font color="Green">E122</font></i></a>, <a class="txt" href="mesfunc5.html#E16:119"><i><font color="Green">E123</font></i></a>, <a class="txt" href="mesfunc5.html#E17:119"><i><font color="Green">E124</font></i></a>, <a class="txt" href="mesfunc5.html#E18:119"><i><font color="Green">E125</font></i></a></i>;<br/>


</div>
