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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E19">A2</font></i>: 
(  <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5" title="CONVFUN1:def.5">CONVFUN1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>case </b></a><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#D13" title="RVSUM_1:def.13">RVSUM_1:def 13</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1_1"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E3:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="binop_2.html#T2" title="BINOP_2:th.2">BINOP_2:2</a>, <a class="ref" href="finsop_1.html#D1" title="FINSOP_1:def.1">FINSOP_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>case </b></a><a NAME="E1:20_1_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#D13" title="RVSUM_1:def.13">RVSUM_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">Q</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> <b> such that </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> holds <br/><font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="finsop_1.html#D1" title="FINSOP_1:def.1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 );<br/><a NAME="E5:20_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E6:20_1_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1_1_2_1"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:20_1_1_2_1"/><i><font color="Green" title="E25">A9</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E2:20_1_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E3:20"><i><font color="Green" title="E19">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E3:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2_1_1_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E4:20_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2_1_1_1"><i><font color="Green" title="E26">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_2_1_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E2:20_1_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:20_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E3:20"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E4:20_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2_1_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="mesfunc3.html#E2:20_1_1_2_1_1_2"><i><font color="Green" title="E27">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>
;<br/><b>hence </b><a NAME="E5:20_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">P</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="mesfunc3.html#E2:20_1_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2_1_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="mesfunc3.html#E3:20_1_1_2_1_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:20_1_1_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="mesfunc3.html#E5:20_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="mesfunc3.html#E6:20_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>);<br/></i><b>hence </b><a NAME="E8:20_1_1_2"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc3.html#E3:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc3.html#E1:20_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="mesfunc3.html#E4:20_1_1_2"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20"/>
 <a href="convfun1.html#K5" title="CONVFUN1:func.5">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">q</font>
 ;<br/>


</div>
