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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span><br/>



<a NAME="E1:55"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></i>;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E35">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<i><font color="Green" title="E36">A4</font></i>: <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#T2" target="_self" title="EUCLID_7:th.2">Th2</a>, <a class="txt" href="#E2:55"><i><font color="Green" title="E35">A3</font></i></a></i>
;<br/>
<a NAME="E4:55"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<b>assume </b><a NAME="E1:55_2"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E2:55_2"/><b>then </b><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">nx</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:55_2"><i><font color="Green" title="E37">A5</font></i></a></i>;<br/>
<a NAME="E4:55_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrixr2.html#T74" title="MATRIXR2:th.74">MATRIXR2:74</a></i>;<br/>
<a NAME="E5:55_2"/><b>then </b><i><font color="Green" title="E39">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">nx</font> &amp; <font color="Maroon" title="c4">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="#E2:55_2"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:55_2"/><i><font color="Green" title="E40">A8</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:55_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1"/>
( <font color="Maroon" title="c4">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c4">nx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1"><b>suppose </b></a><a NAME="E1:55_2_1_1"/><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c4">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="#E5:55_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="#E6:55_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="matrixr2.html#T75" title="MATRIXR2:th.75">MATRIXR2:75</a></i>
<br/>.= 
1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2" title="RVSUM_1:def.2">RVSUM_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="#E5:55_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="#E1:55_2_1_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="matrixr2.html#T75" title="MATRIXR2:th.75">MATRIXR2:75</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_2"><b>suppose </b></a><a NAME="E1:55_2_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c4">nx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<a href="rvsum_1.html#K2" title="RVSUM_1:func.2">sqrreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E5:55_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="#E6:55_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2" title="RVSUM_1:def.2">RVSUM_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="#E5:55_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="#E1:55_2_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
 <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="#E3:55"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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