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<span class="kw">let </span><font color="Maroon" title="c1">cn</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <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> 2</span>)</span> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</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="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</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> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c2">p</font> as   <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> 2</span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">qz</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> as   <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> 2</span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1"/>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or <font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_1"/><span class="lab"><font color="Green" title="E80">A1</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_4.html#T56" target="_self" title="JGRAPH_4:th.56">Th56</a></span>;<br/><a NAME="E3:90_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A3</font></span>: 
<font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a NAME="E4:90_1_1"/><span class="lab"><font color="Green" title="E83">A4</font></span>: 
<font color="Maroon" title="c6">qz</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1_1"><span class="lab"><font color="Green" title="E81">A2</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a NAME="E5:90_1_1"/><span class="lab"><font color="Green" title="E84">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E6:90_1_1"/><span class="lab"><font color="Green" title="E85">A6</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>;<br/><a NAME="E7:90_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="jgraph_2.html#T11" title="JGRAPH_2:th.11">JGRAPH_2:11</a>, <a class="ref" href="toprns_1.html#T25" title="TOPRNS_1:th.25">TOPRNS_1:25</a></span>;<br/><a NAME="E8:90_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A7</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></span>;<br/><a NAME="E9:90_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/><a NAME="E10:90_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E11:90_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_1_1"><span class="lab"><font color="Green" title="E85">A6</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E12:90_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1_1"><span class="lab"><font color="Green" title="E86">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E13:90_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>;<br/><a NAME="E14:90_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T121" title="SQUARE_1:th.121">SQUARE_1:121</a></span>;<br/><a NAME="E15:90_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A8</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_1_1"/>
( 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_1"/><span class="lab"><font color="Green" title="E88">A9</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><span class="lab"><font color="Green" title="E89">A10</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_1"><span class="lab"><font color="Green" title="E88">A9</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E3:90_1_1_1_1"/><span class="kw">then </span>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font> = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="txt" href="#E2:90_1_1_1_1"><span class="lab"><font color="Green" title="E89">A10</font></span></a>, <a class="ref" href="jgraph_4.html#T56" target="_self" title="JGRAPH_4:th.56">Th56</a>, <a class="ref" href="square_1.html#T83" title="SQUARE_1:th.83">SQUARE_1:83</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>

;<br/><a NAME="E5:90_1_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:90_1_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_2"/><span class="lab"><font color="Green" title="E88">A11</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_1_1_2_1"/>
( 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_2"><span class="lab"><font color="Green" title="E88">A11</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E89">A12</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_1_1_2_1_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:90_1_1"><span class="lab"><font color="Green" title="E87">A8</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><a NAME="E3:90_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E89">A12</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E4:90_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E89">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" title="XCMPLX_1:th.198">XCMPLX_1:198</a></span>;<br/><a NAME="E5:90_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1_1"><span class="lab"><font color="Green" title="E84">A5</font></span></a>, <a class="txt" href="#E1:90_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E89">A12</font></span></a>, <a class="ref" href="square_1.html#T119" title="SQUARE_1:th.119">SQUARE_1:119</a></span>;<br/><a NAME="E6:90_1_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E7:90_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A13</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></span>;<br/><span class="lab"><font color="Green" title="E91">A14</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_1"><span class="lab"><font color="Green" title="E82">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E90">A13</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">qz</font></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>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1_1"><span class="lab"><font color="Green" title="E83">A4</font></span></a>, <a class="txt" href="#E8:90_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E91">A14</font></span></a></span>
;<br/><a NAME="E10:90_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">qz</font></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> </span>)</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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/><span class="kw">hence </span><a NAME="E11:90_1_1_1_2_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E89">A15</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_1_1_2_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E3:90_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1_1"><span class="lab"><font color="Green" title="E86">A7</font></span></a>, <a class="txt" href="#E6:90_1_1"><span class="lab"><font color="Green" title="E85">A6</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/><a NAME="E4:90_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1_1"><span class="lab"><font color="Green" title="E86">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E5:90_1_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>;<br/><a NAME="E6:90_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A16</font></span>: 
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T122" title="SQUARE_1:th.122">SQUARE_1:122</a></span>;<br/><a NAME="E7:90_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><span class="kw">hence </span><a NAME="E8:90_1_1_1_2_1_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E89">A15</font></span></a>, <a class="txt" href="#E6:90_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E90">A16</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_2"/><span class="lab"><font color="Green" title="E80">A17</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T11" title="JGRAPH_2:th.11">JGRAPH_2:11</a>, <a class="ref" href="toprns_1.html#T25" title="TOPRNS_1:th.25">TOPRNS_1:25</a></span>;<br/><a NAME="E3:90_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A18</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></span>;<br/><a NAME="E4:90_1_2"/><span class="lab"><font color="Green" title="E82">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></span>;<br/><a NAME="E5:90_1_2"/><span class="lab"><font color="Green" title="E83">A20</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>;<br/><a NAME="E6:90_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/><a NAME="E7:90_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:90_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1_2"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E9:90_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_2"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E10:90_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>;<br/><a NAME="E11:90_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T121" title="SQUARE_1:th.121">SQUARE_1:121</a></span>;<br/><a NAME="E12:90_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A21</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E13:90_1_2"/><span class="lab"><font color="Green" title="E85">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="jgraph_4.html#T57" target="_self" title="JGRAPH_4:th.57">Th57</a></span>;<br/><a NAME="E14:90_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A23</font></span>: 
<font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a NAME="E15:90_1_2"/><span class="lab"><font color="Green" title="E87">A24</font></span>: 
<font color="Maroon" title="c6">qz</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:90_1_2"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_2_1"/>
( 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_2_1_1"/><span class="lab"><font color="Green" title="E88">A25</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2_1_1"><span class="lab"><font color="Green" title="E88">A25</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E3:90_1_2_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:90_1_2"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="square_1.html#T83" title="SQUARE_1:th.83">SQUARE_1:83</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:90_1_2_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_2_1_2"/><span class="lab"><font color="Green" title="E88">A26</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_2_1_2_1"/>
( 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2_1_2"><span class="lab"><font color="Green" title="E88">A26</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E89">A27</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90_1_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E3:90_1_2_1_2_1_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E89">A27</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" title="XCMPLX_1:th.198">XCMPLX_1:198</a></span>;<br/><a NAME="E4:90_1_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1_2"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="txt" href="#E1:90_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E89">A27</font></span></a>, <a class="ref" href="square_1.html#T119" title="SQUARE_1:th.119">SQUARE_1:119</a></span>;<br/><a NAME="E5:90_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A28</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><span class="lab"><font color="Green" title="E91">A29</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:90_1_2"><span class="lab"><font color="Green" title="E86">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p0"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E90">A28</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">qz</font></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>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">qz</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:90_1_2"><span class="lab"><font color="Green" title="E87">A24</font></span></a>, <a class="txt" href="#E6:90_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E91">A29</font></span></a></span>
;<br/><a NAME="E8:90_1_2_1_2_1_1"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">qz</font></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> </span>)</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="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/><span class="kw">hence </span><a NAME="E9:90_1_2_1_2_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E89">A30</font></span>: 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:90_1_2_1_2_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E3:90_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_2"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="txt" href="#E5:90_1_2"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/><a NAME="E4:90_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></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> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1_2"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E5:90_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></span>;<br/><a NAME="E6:90_1_2_1_2_1_2"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T122" title="SQUARE_1:th.122">SQUARE_1:122</a></span>;<br/><a NAME="E7:90_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">cn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E8:90_1_2_1_2_1_2"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">cn</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E89">A30</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E9:90_1_2_1_2_1_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E7:90_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E90">A31</font></span></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_3"><span class="kw">suppose </span></a><a NAME="E1:90_1_3"/>
<font color="Maroon" title="c5">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:90_1_3"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font></span>)</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="c2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_4.html#T56" target="_self" title="JGRAPH_4:th.56">Th56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
