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

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">h</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:13"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<i><font color="Green">E51</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E6:13"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:13_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:13_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:13_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:13_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_2"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:13_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i>, , , , , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:13_2_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:13_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:13_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:13_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:13_2_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:13_2_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:13_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E12:13_2_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, , , , , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:13"/>
 <a href="rvsum_1.html#K11">sqr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
