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<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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:101"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">xr</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 1 ;<br/>
<a NAME="E3:101"/>
<font color="Maroon">q</font> is    <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#T25">EUCLID:25</a></i>;<br/>
<a NAME="E4:101"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E5:101"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E56</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">xr</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <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">xr</font></span>)</span></span>)</span>

;<br/>
<a NAME="E7:101"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E8:101"/><b>then </b>
<font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:101"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E10:101"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E11:101"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E12:101"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E14:101"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>


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