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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:17"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="absvalue.html#T1">ABSVALUE:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E3:17_1_1_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sx</font> =  <a href="euclid.html#K11">sqr</a> <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:17_2"/>
<font color="Maroon">sx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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/><b>hence </b><a NAME="E3:17_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sx</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:17"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E8:17"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E9:17"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E10:17"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E11:17"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:17"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">x</font></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>;<br/>


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