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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y1</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">z2</font> = <font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font>;<br/>
<i><font color="Green" title="E12">A2</font></i>: <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D5" title="EUCLID:def.5">EUCLID:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D5" title="EUCLID:def.5">EUCLID:def 5</a></i>
;<br/>
<a NAME="E3:12"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E5:12"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></i>;<br/>
<a NAME="E6:12"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">v1</font> =  <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">v2</font> =  <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">w1</font> = <font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c11">w2</font> = <font color="Maroon" title="c5">x2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1" title="EUCLID:def.1">EUCLID:def 1</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E15">A5</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p2">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T120" title="RVSUM_1:th.120">RVSUM_1:120</a></i>;<br/>
<i><font color="Green" title="E16">A6</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></i>
;<br/>
<a NAME="E10:12"/><i><font color="Green" title="E17">A7</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span>
 
;<br/>
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></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#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">w1</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T98" title="RVSUM_1:th.98">RVSUM_1:98</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#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T99" title="RVSUM_1:th.99">RVSUM_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid_2.html#E10:12"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c10">w1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T62" title="RVSUM_1:th.62">RVSUM_1:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T63" title="RVSUM_1:th.63">RVSUM_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T63" title="RVSUM_1:th.63">RVSUM_1:63</a>, <a class="txt" href="euclid_2.html#E9:12"><i><font color="Green" title="E16">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></i>
<br/>.= 
4 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span>

;<br/>
<b>then </b><span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid_2.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="euclid_2.html#E4:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="euclid_2.html#E6:12"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="euclid_2.html#E8:12"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c10">w1</font>,<font color="Maroon" title="c11">w2</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E13:12"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 4</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euclid_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>


</div>
