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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E86">A1</font></i>: 
<font color="Maroon" title="c3">q</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="c2">r</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">xr</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</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 ;<br/>
<a NAME="E3:96"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E4:96"/><b>then </b>
<font color="Maroon" title="c4">xr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">xr</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E5:96"/><b>then </b><i><font color="Green" title="E87">A2</font></i>: 
 <a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Maroon" title="c3">q</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">xr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="ref" href="jordan2b.html#D1" title="JORDAN2B:def.1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E88">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c4">xr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E7:96"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c4">xr</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Maroon" title="c3">q</font>,1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:96"><i><font color="Green" title="E87">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T79" title="RVSUM_1:th.79">RVSUM_1:79</a></i>;<br/>
<a NAME="E8:96"/><b>then </b><i><font color="Green" title="E89">A4</font></i>: 
 <a href="euclid.html#K11" title="EUCLID:func.11">sqr</a> <font color="Maroon" title="c4">xr</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"><span class="p3">(<span class="default"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Maroon" title="c3">q</font>,1</span>)</span> <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="txt" href="jordan2c.html#E6:96"><i><font color="Green" title="E88">A3</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Maroon" title="c3">q</font>,1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1" title="JORDAN2B:func.1">Proj</a> <font color="Maroon" title="c3">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158" title="COMPLEX1:th.158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:96"><i><font color="Green" title="E87">A2</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E10:96"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">q</font></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="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:96"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="jordan2c.html#E8:96"><i><font color="Green" title="E89">A4</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/>


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