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

<b>let </b><font color="Maroon">cn</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:96"/><i><font color="Green">E34</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">cn</font> &amp; <font color="Maroon">cn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">cn</font> &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">cn</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> )
 ;<br/>

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:96"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K5" target="_self">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K5" target="_self">-FanMorphN</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q2</font> )
 ;<br/>

<a NAME="E3:96"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</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">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</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">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:96"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</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">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</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">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:96"/>
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</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">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</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">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:96"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</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">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K4">*</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">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:96"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="jgraph_2.html#T11">JGRAPH_2:11</a></i>;<br/>
<a NAME="E8:96"/><i><font color="Green">E41</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:96"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E10:96"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E11:96"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">cn</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:96"/>
1 <a href="real_1.html#K3">+</a> <font color="Maroon">cn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:96"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, , , <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>


</div>
