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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:40"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> 1
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:40"/><i><font color="Green">E34</font></i>: 
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finsop_1.html#T2">FINSOP_1:2</a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:40"/><i><font color="Green">E35</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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:40_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>hence </b><a NAME="E2:40_1"/>
( <font color="Maroon">c<sub>5</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>1</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:40"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<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> st <font color="Olive">b<sub>2</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>1</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <font color="Olive">b<sub>2</sub></font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="matrixc1.html#E6:40"><i><font color="Green">E35</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:40"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</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>1</sub></font></span>)</span> &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>1</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 ;<br/>
<a NAME="E10:40"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green">E38</font></i>: 
1 <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>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:40"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E40</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:40_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E3:40_2"/><b>then </b><i><font color="Green">E41</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:40_2"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E3:40_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E5:40_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:40_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</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>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E3:40_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="matrixc1.html#E1:40_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E7:40_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <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>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E5:40_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrixc1.html#E4:40_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:40"><i><font color="Green">E33</font></i></a>, <a class="txt" href="matrixc1.html#E1:40_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E6:40_2"><i><font color="Green">E44</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:40_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:40"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<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>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E41</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> 1 = 
<font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> 1
<b>by </b><i><a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E3:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrixc1.html#E11:40"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> 1

;<br/>
<a NAME="E15:40"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E12:40"><i><font color="Green">E39</font></i></a></i>;<br/>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E5:40"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrixc1.html#E13:40"><i><font color="Green">E40</font></i></a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E18:40"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E19:40"/><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E10:40"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E20:40"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E18:40"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrixc1.html#E19:40"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E21:40"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:40_5"/><i><font color="Green">E48</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:40_5"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:40_5"/><b>then </b><i><font color="Green">E49</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:40_5"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="matrixc1.html#E9:40"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:40_5"/><i><font color="Green">E51</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:40_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E10:40"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E7:40_5"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E18:40"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrixc1.html#E5:40_5"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E8:40_5"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:40_5"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrixc1.html#E4:40_5"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrixc1.html#E18:40"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E9:40_5"/>
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complsp2.html#K10">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#T44">COMPLSP2:44</a>, <a class="txt" href="matrixc1.html#E8:40_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:40_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:40_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrixc1.html#E15:40"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrixc1.html#E7:40_5"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:40"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" target="_self">FR2FC</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="matrixc1.html#E18:40"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrixc1.html#E10:40"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:40"/>
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finsop_1.html#T3">FINSOP_1:3</a>, <a class="txt" href="matrixc1.html#E20:40"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrixc1.html#E21:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrixc1.html#E16:40"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrixc1.html#E14:40"><i><font color="Green">E41</font></i></a></i>;<br/>


</div>
