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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>set </b><font color="Maroon" title="c2">p</font> = <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">w</font> = <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> as    <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> 1 <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E2:10"/><i><font color="Green" title="E11">A1</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
;<br/>
<a NAME="E3:10"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c3">w</font></span>)</span>
 
<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="E4:10"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></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>  <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> <font color="Maroon" title="c3">w</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E2:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26" title="TOPRNS_1:th.26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E6:10"/>
<font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#D2" title="JORDAN2B:def.2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
 <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T81" title="RVSUM_1:th.81">RVSUM_1:81</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/>
<b>then </b><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:10"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="ref" href="complex1.html#T158" title="COMPLEX1:th.158">COMPLEX1:158</a></i>
;<br/>
<b>hence </b><a NAME="E10:10"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c1">r</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c1">r</font>
 ;<br/>


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