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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  holds <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:91"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:91_1"/>
(  <a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  &amp;  <a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/>
<b>hence </b><a NAME="E2:91_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T102" target="_self">Th102</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:91"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:91_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  holds <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:91_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:91_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<a NAME="E6:91_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:91_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E7:91_2"/><b>then </b><i><font color="Green">E64</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<i><font color="Green">E65</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" target="_self">Th117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T84" target="_self">Th84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" target="_self">Th117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T84" target="_self">Th84</a></i>
;<br/>
 <a href="rvsum_1.html#K13" target="_self">mlt</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseqop.html#T11">FINSEQOP:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/>
<a NAME="E10:91_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" target="_self">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" target="_self">Th104</a></i>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> = 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" target="_self">Th117</a></i>
<br/>.= 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" target="_self">Th71</a></i>
<br/>.= 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94" target="_self">Th94</a></i>
<br/>.= 
2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94" target="_self">Th94</a></i>
;<br/>
<i><font color="Green">E68</font></i>:  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E11:91_2"><i><font color="Green">E67</font></i></a></i>
;<br/>
<i><font color="Green">E69</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" target="_self">Th117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="rvsum_1.html#K8" target="_self">-</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T120" target="_self">Th120</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="rvsum_1.html#K8" target="_self">-</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119" target="_self">Th119</a></i>
;<br/>
<a NAME="E14:91_2"/>
for <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> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  holds  <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11" target="_self">sqr</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>



 <a href="rvsum_1.html#K11" target="_self">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" target="_self">sqrreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" target="_self">sqrreal</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseqop.html#T9">FINSEQOP:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E2:91_2_6"/>
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11" target="_self">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T104" target="_self">Th104</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:91_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:91_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rvsum_1.html#E5:91_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:91_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rvsum_1.html#E5:91_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="rvsum_1.html#E10:91_2"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E8:91_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rvsum_1.html#E12:91_2"><i><font color="Green">E68</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rvsum_1.html#K8" target="_self">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E13:91_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="rvsum_1.html#T99" target="_self">Th99</a></i>
;<br/>
<a NAME="E17:91_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rvsum_1.html#K8" target="_self">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rvsum_1.html#K10" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" target="_self">Th116</a></i>;<br/>
<a NAME="E18:91_2"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E7:91_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rvsum_1.html#E16:91_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E19:91_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E20:91_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:91"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="rvsum_1.html#E1:91"><i><font color="Green">E59</font></i></a>, <a class="txt" href="rvsum_1.html#E2:91"><i><font color="Green">E60</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:91"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
