<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">sn</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</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>  st  <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> <font color="Maroon" title="c1">sn</font> &amp; <font color="Maroon" title="c1">sn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b1">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  &amp; <font color="Olive" title="b2">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b1">q1</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b2">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> holds <br/> for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</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>  st <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q1</font> &amp; <font color="Olive" title="b4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b3">p1</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> <span class="p1">(<span class="default"><font color="Olive" title="b4">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b4">p2</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">q1</font>, <font color="Maroon" title="c3">q2</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> <font color="Maroon" title="c1">sn</font> &amp; <font color="Maroon" title="c1">sn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  &amp; <font color="Maroon" title="c3">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b1">p1</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b2">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:128"/><span class="lab"><font color="Green" title="E120">A1</font></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> <font color="Maroon" title="c1">sn</font>
 <span class="kw">and </span><br/><a NAME="E2:128"/><span class="lab"><font color="Green" title="E121">A2</font></span>: 
<font color="Maroon" title="c1">sn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">and </span><br/><a NAME="E3:128"/><span class="lab"><font color="Green" title="E122">A3</font></span>: 
<font color="Maroon" title="c2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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">and </span><br/><a NAME="E4:128"/><span class="lab"><font color="Green" title="E123">A4</font></span>: 
<font color="Maroon" title="c3">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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">and </span><br/><a NAME="E5:128"/><span class="lab"><font color="Green" title="E124">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b1">p1</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="Olive" title="b2">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q1</font> &amp; <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:128"/><span class="lab"><font color="Green" title="E125">A6</font></span>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q1</font>
 <span class="kw">and </span><br/><a NAME="E7:128"/><span class="lab"><font color="Green" title="E126">A7</font></span>: 
<font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">sn</font> <a href="jgraph_4.html#K7" title="JGRAPH_4:func.7">-FanMorphE</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_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="E1:128_1"/>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1"><span class="kw">suppose </span></a><a NAME="E1:128_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="E2:128_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="#E2:128"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="txt" href="#E4:128"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E5:128"><span class="lab"><font color="Green" title="E124">A5</font></span></a>, <a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E125">A6</font></span></a>, <a class="txt" href="#E7:128"><span class="lab"><font color="Green" title="E126">A7</font></span></a>, <a class="ref" href="jgraph_4.html#T108" target="_self" title="JGRAPH_4:th.108">Th115</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="128_1_2"><span class="kw">suppose </span></a><a NAME="E1:128_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="E2:128_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="#E5:128"><span class="lab"><font color="Green" title="E124">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="128_1_3"><span class="kw">suppose </span></a><a NAME="E1:128_1_3"/><span class="lab"><font color="Green" title="E127">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/><div class="add"><a NAME="E2:128_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="#E2:128"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <a class="txt" href="#E4:128"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E7:128"><span class="lab"><font color="Green" title="E126">A7</font></span></a>, <a class="ref" href="jgraph_4.html#T106" target="_self" title="JGRAPH_4:th.106">Th113</a></span>;<br/><a NAME="E3:128_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:128_1_3"/>
<font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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="#E1:128"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E125">A6</font></span></a>, <a class="txt" href="#E1:128_1_3"><span class="lab"><font color="Green" title="E127">A8</font></span></a>, <a class="ref" href="jgraph_4.html#T107" target="_self" title="JGRAPH_4:th.107">Th114</a></span>;<br/><span class="kw">hence </span><a NAME="E5:128_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="#E3:128_1_3"><span class="lab"><font color="Green" title="E128">A9</font></span></a>, <a class="txt" href="jgraph_4.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="jgraph_2.html#T3" title="JGRAPH_2:th.3">JGRAPH_2:3</a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="128_1_4"><span class="kw">suppose </span></a><a NAME="E1:128_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">q1</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">sn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c3">q2</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">sn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="E2:128_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c4">p1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</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="#E1:128"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="txt" href="#E4:128"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="txt" href="#E5:128"><span class="lab"><font color="Green" title="E124">A5</font></span></a>, <a class="txt" href="#E6:128"><span class="lab"><font color="Green" title="E125">A6</font></span></a>, <a class="txt" href="#E7:128"><span class="lab"><font color="Green" title="E126">A7</font></span></a>, <a class="ref" href="jgraph_4.html#T109" target="_self" title="JGRAPH_4:th.109">Th116</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
