<?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>, <font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E7</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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green">E8</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>and </b><br/><a NAME="E5:6"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green">E10</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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <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="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E8:6"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green">E12</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E10:6"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:6"/><i><font color="Green">E14</font></i>: 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6"><i><font color="Green">E7</font></i></a>, <a class="ref" href="mesfunc4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E12:6"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:6"><i><font color="Green">E5</font></i></a>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<a NAME="E13:6"/>
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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:6_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mesfunc4.html#E12:6"><i><font color="Green">E15</font></i></a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E3:6_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/>
<a NAME="E4:6_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <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>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <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>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="extreal1.html#T27">EXTREAL1:27</a>, <a class="ref" href="extreal1.html#T28">EXTREAL1:28</a></i>;<br/>
<a NAME="E5:6_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="mesfunc4.html#E6:6"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T6">XXREAL_0:6</a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>
<a NAME="E7:6_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T59">EXTREAL2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:6"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E15:6"/><i><font color="Green">E17</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</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="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:6_2"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:6_2"/><i><font color="Green">E19</font></i>: 
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>, <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>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <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>3</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6"><i><font color="Green">E5</font></i></a>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<a NAME="E4:6_2"/><i><font color="Green">E20</font></i>: 
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>, <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>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><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/><b>let </b><font color="Maroon">c<sub>12</sub></font>, <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="E1:6_2_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<a NAME="E2:6_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:6_2_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mesfunc4.html#E1:6_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:6_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <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>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <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>13</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E6:6"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:6_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_2"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E5:6_2"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></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>, <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>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mesfunc4.html#E4:6_2"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:6"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</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="mesfunc4.html#E14:6"><i><font color="Green">E16</font></i></a>, <a class="txt" href="mesfunc4.html#E15:6"><i><font color="Green">E17</font></i></a>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<a NAME="E17:6"/><i><font color="Green">E19</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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><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="E1:6_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:6_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E3:6_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <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>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mesfunc4.html#E6:6"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</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="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E20:6"/><i><font color="Green">E20</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.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="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E21:6"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="mesfunc4.html#E3:6_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:6_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E20:6"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:6_4"/><b>then </b>
<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>11</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E20:6"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E23:6"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5"><b>proof </b></a><div class="add">

<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>  = <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:6_5"/><i><font color="Green">E21</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</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="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E3:6_5"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:6_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_5_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_5_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="mesfunc4.html#E3:6_5_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E5:6_5_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:6_5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_5_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:6_5_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>thus </b><a NAME="E6:6_5"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</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">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:6_5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E25:6"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E26:6"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Maroon">c<sub>14</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">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>
<a NAME="E27:6"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6"><b>proof </b></a><div class="add">

<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>  = <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:6_6"/><i><font color="Green">E23</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</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="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E3:6_6"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:6_6"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_6_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_6_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="mesfunc4.html#E3:6_6_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:6_6_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:6_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_6"><i><font color="Green">E24</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_6_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E6:6_6"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>16</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">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:6_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mesfunc4.html#E3:6_6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E29:6"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E30:6"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</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">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>
<a NAME="E31:6"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_7"><b>proof </b></a><div class="add">

<i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/>
<i><font color="Green">E27</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E25:6"><i><font color="Green">E21</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_7_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:6_7_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:6_7_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E3:6_7_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_7_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E4:6_7_3"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_7"><i><font color="Green">E26</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6_7_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:6_7_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_7_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mesfunc4.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc4.html#E25:6"><i><font color="Green">E21</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <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>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:6_7_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="mesfunc4.html#E6:6"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:6_7_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mesfunc4.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="txt" href="mesfunc4.html#E1:6_7_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:6_7_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="mesfunc4.html#E26:6"><i><font color="Green">E22</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_7"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_7"><i><font color="Green">E26</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6_7"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E26</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E29:6"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E9:6"><i><font color="Green">E12</font></i></a></i>
;<br/>
<a NAME="E33:6"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</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>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="6_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:6_9"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:6_9"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>15</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E30:6"><i><font color="Green">E24</font></i></a>, <a class="txt" href="mesfunc4.html#E1:6_9"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:6_9"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mesfunc4.html#E10:6"><i><font color="Green">E13</font></i></a>, <a class="txt" href="mesfunc4.html#E29:6"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="extreal1.html#T23">EXTREAL1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</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>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E26:6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mesfunc4.html#E1:6_9"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mesfunc4.html#E25:6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6_9"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mesfunc4.html#E32:6"><i><font color="Green">E26</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E16:6"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mesfunc4.html#E2:6"><i><font color="Green">E6</font></i></a>, <a class="txt" href="mesfunc4.html#E5:6"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mesfunc4.html#E17:6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mesfunc4.html#E31:6"><i><font color="Green">E25</font></i></a>, <a class="txt" href="mesfunc4.html#E32:6"><i><font color="Green">E26</font></i></a>, <a class="txt" href="mesfunc4.html#E33:6"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mesfunc4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mesfunc4.html#E29:6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mesfunc4.html#E30:6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="mesfunc3.html#T10">MESFUNC3:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="mesfunc3.html#K1">integral</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E11:6"><i><font color="Green">E14</font></i></a></i>
;<br/>


</div>
