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

<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>, <font color="Maroon">p3</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">p4</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>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</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="E1:62"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>

<a NAME="E2:62"/><b>then </b>
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:62"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:62"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E7:62"/>
<span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:62"/><b>then </b><i><font color="Green">E33</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <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><b> such that </b><br/><a NAME="E10:62"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">p11</font> being   <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><b> such that </b><br/><a NAME="E12:62"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p11</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">p22</font> being   <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><b> such that </b><br/><a NAME="E14:62"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p22</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">p33</font> being   <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><b> such that </b><br/><a NAME="E16:62"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">p33</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p33</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E17:62"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:62"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:62"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E20:62"/><b>then </b>
<font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T7">JGRAPH_2:7</a></i>;<br/>
<a NAME="E21:62"/><b>then </b><i><font color="Green">E49</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E22:62"/><b>then </b><i><font color="Green">E73</font></i>: 
<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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E23:62"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/>
<a NAME="E24:62"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E25:62"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T5">JGRAPH_4:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E27:62"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_2.html#R2">=</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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="jgraph_4.html#K3">-FanMorphW</a>  &amp; <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i><a class="ref" href="jgraph_4.html#T48">JGRAPH_4:48</a></i>;<br/>
<a NAME="E28:62"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">q</font> being  <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> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></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="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="jgraph_4.html#T40">JGRAPH_4:40</a></i>;<br/>
<a NAME="E29:62"/><i><font color="Green">E79</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">q11</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>;<br/>
<b>set </b><font color="Maroon">q22</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>;<br/>
<b>set </b><font color="Maroon">q33</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>;<br/>
<b>set </b><font color="Maroon">q44</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>;<br/>
<a NAME="E30:62"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="jgraph_4.html#T40">JGRAPH_4:40</a></i>;<br/>
<a NAME="E31:62"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E32:62"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="jgraph_4.html#T40">JGRAPH_4:40</a></i>;<br/>
<a NAME="E33:62"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E34:62"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="jgraph_4.html#T40">JGRAPH_4:40</a></i>;<br/>
<a NAME="E35:62"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E36:62"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="jgraph_4.html#T40">JGRAPH_4:40</a></i>;<br/>
<a NAME="E37:62"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E38:62"/><i><font color="Green">E90</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<a NAME="E39:62"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E40:62"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E41:62"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E42:62"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:62"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E43:62"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:62"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E44:62"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E45:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E32:62"><i><font color="Green">E83</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E46:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E47:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E48:62"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E49:62"/><b>then </b><i><font color="Green">E99</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:62"/><i><font color="Green">E100</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/>
<a NAME="E51:62"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</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>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jgraph_5.html#E36:62"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E37:62"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_2.html#T6">JGRAPH_2:6</a></i>;<br/>
<a NAME="E52:62"/><b>then </b><i><font color="Green">E102</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T50">JGRAPH_4:50</a></i>;<br/>
<a NAME="E53:62"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E54:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E32:62"><i><font color="Green">E83</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E55:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E56:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E57:62"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E58:62"/><b>then </b><i><font color="Green">E103</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:62"/><i><font color="Green">E105</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/>
<a NAME="E60:62"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="jgraph_2.html#T6">JGRAPH_2:6</a></i>;<br/>
<a NAME="E61:62"/><b>then </b><i><font color="Green">E107</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T50">JGRAPH_4:50</a></i>;<br/>
<a NAME="E62:62"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:62"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E63:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E32:62"><i><font color="Green">E83</font></i></a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E64:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E65:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E66:62"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E67:62"/><b>then </b><i><font color="Green">E136</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E68:62"/><i><font color="Green">E138</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/>
<a NAME="E69:62"/><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K22">`2</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">p3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , , <a class="txt" href="jgraph_5.html#E41:62"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jgraph_2.html#T6">JGRAPH_2:6</a></i>;<br/>
<a NAME="E70:62"/><b>then </b><i><font color="Green">E140</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T50">JGRAPH_4:50</a></i>;<br/>
<a NAME="E71:62"/><i><font color="Green">E141</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/>
<a NAME="E72:62"/>
<span class="p1">(<span class="default"><font color="Maroon">p4</font> <a href="euclid.html#K22">`2</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">p4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E50:62"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_2.html#T6">JGRAPH_2:6</a></i>;<br/>
<a NAME="E73:62"/><b>then </b><i><font color="Green">E142</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="txt" href="jgraph_5.html#E38:62"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_5.html#E39:62"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_5.html#E42:62"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_4.html#T50">JGRAPH_4:50</a></i>;<br/>
<a NAME="E74:62"/>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</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#K22">`2</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> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E75:62"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T53">JGRAPH_4:53</a></i>;<br/>
<a NAME="E76:62"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E25:62"><i><font color="Green">E75</font></i></a>, , <a class="txt" href="jgraph_5.html#E27:62"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E28:62"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="jgraph_5.html#E40:62"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E77:62"/>
( <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</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="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K22">`2</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">p3</font></span><a href="toprns_1.html#K5">.|</a></span> or <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E78:62"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> or <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T53">JGRAPH_4:53</a></i>;<br/>
<a NAME="E79:62"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E27:62"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jgraph_5.html#E28:62"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_5.html#E29:62"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E30:62"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jgraph_5.html#E40:62"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_5.html#E49:62"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E80:62"/>
( <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K22">`2</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">p3</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">p4</font> <a href="euclid.html#K22">`2</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">p4</font></span><a href="toprns_1.html#K5">.|</a></span> or <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="txt" href="jgraph_5.html#E16:62"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E81:62"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> or <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jgraph_4.html#T53">JGRAPH_4:53</a></i>;<br/>
<a NAME="E82:62"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_5.html#E29:62"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E30:62"><i><font color="Green">E80</font></i></a>, , <a class="txt" href="jgraph_5.html#E31:62"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E51:62"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E83:62"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">q3</font>, <font color="Olive">q4</font> being  <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> st <br/>( <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">q</font> being  <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> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></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="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> &amp; <font color="Olive">q3</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> &amp; <font color="Olive">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q2</font>,<font color="Olive">q3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q3</font>,<font color="Olive">q4</font>,<font color="Maroon">P</font> )
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E22:62"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E51:62"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_5.html#E52:62"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_5.html#E58:62"><i><font color="Green">E103</font></i></a></i>;<br/>


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