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

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p4</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>;<br/><b>let </b><font color="Maroon" title="c5">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E61">A1</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span>}</span>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:63"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E62">A2</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span>}</span>  &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E63">A3</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">p11</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><b> such that </b><br/><a NAME="E7:63"/><i><font color="Green" title="E64">A4</font></i>: 
( <font color="Maroon" title="c7">p11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">p11</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:63"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">p22</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><b> such that </b><br/><a NAME="E9:63"/><i><font color="Green" title="E65">A5</font></i>: 
( <font color="Maroon" title="c8">p22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">p22</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:63"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">p33</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><b> such that </b><br/><a NAME="E11:63"/><i><font color="Green" title="E66">A6</font></i>: 
( <font color="Maroon" title="c9">p33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">p33</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:63"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>
<a NAME="E12:63"/><i><font color="Green" title="E67">A7</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="jgraph_5.html#T3" target="_self" title="JGRAPH_5:th.3">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:63_1"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 ;<br/><b>hence </b><a NAME="E4:63_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:63"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:63"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E16:63"/><i><font color="Green" title="E68">A8</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp; <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">r1</font> = <font color="Maroon" title="c10">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E18:63"/><i><font color="Green" title="E69">A9</font></i>: 
(  <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="c11">r1</font> &amp; <font color="Maroon" title="c11">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E12:63"><i><font color="Green" title="E67">A7</font></i></a>, <a class="txt" href="jgraph_5.html#E16:63"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E20:63"/><i><font color="Green" title="E70">A10</font></i>: 
( <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">r1</font> <a href="jgraph_4.html#K5" title="JGRAPH_4:func.5">-FanMorphN</a>  &amp; <font color="Maroon" title="c12">f1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  )
 <b>by </b><i><a class="ref" href="jgraph_4.html#T81" title="JGRAPH_4:th.81">JGRAPH_4:81</a></i>;<br/>
<a NAME="E21:63"/><i><font color="Green" title="E71">A11</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T73" title="JGRAPH_4:th.73">JGRAPH_4:73</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">q11</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>;<br/>
<b>set </b><font color="Maroon" title="c14">q22</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>;<br/>
<b>set </b><font color="Maroon" title="c15">q33</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>;<br/>
<b>set </b><font color="Maroon" title="c16">q44</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>;<br/>
<a NAME="E22:63"/><i><font color="Green" title="E72">A12</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T73" title="JGRAPH_4:th.73">JGRAPH_4:73</a></i>;<br/>
<a NAME="E23:63"/><b>then </b><i><font color="Green" title="E73">A13</font></i>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>
<a NAME="E24:63"/><i><font color="Green" title="E74">A14</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T73" title="JGRAPH_4:th.73">JGRAPH_4:73</a></i>;<br/>
<a NAME="E25:63"/><b>then </b><i><font color="Green" title="E75">A15</font></i>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>
<a NAME="E26:63"/><i><font color="Green" title="E76">A16</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T73" title="JGRAPH_4:th.73">JGRAPH_4:73</a></i>;<br/>
<a NAME="E27:63"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>
<a NAME="E28:63"/><i><font color="Green" title="E78">A18</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T73" title="JGRAPH_4:th.73">JGRAPH_4:73</a></i>;<br/>
<a NAME="E29:63"/><b>then </b><i><font color="Green" title="E79">A19</font></i>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>
<a NAME="E30:63"/><i><font color="Green" title="E80">A20</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E31:63"/><i><font color="Green" title="E81">A21</font></i>: 
( <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E32:63"/><i><font color="Green" title="E82">A22</font></i>: 
( <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E33:63"/><b>then </b><i><font color="Green" title="E83">A23</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E16:63"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:63"/><b>then </b><i><font color="Green" title="E84">A24</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E31:63"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:63"/><i><font color="Green" title="E85">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p2</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="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E31:63"><i><font color="Green" title="E81">A21</font></i></a>, <a class="txt" href="jgraph_5.html#E33:63"><i><font color="Green" title="E83">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:63"/><b>then </b><i><font color="Green" title="E86">A26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T23" target="_self" title="JGRAPH_5:th.23">Th23</a></i>;<br/>
<a NAME="E37:63"/><i><font color="Green" title="E87">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">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> <font color="Maroon" title="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E30:63"><i><font color="Green" title="E80">A20</font></i></a>, <a class="txt" href="jgraph_5.html#E34:63"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E38:63"/><b>then </b><i><font color="Green" title="E88">A28</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T23" target="_self" title="JGRAPH_5:th.23">Th23</a></i>;<br/>
<a NAME="E39:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">p4</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="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E16:63"><i><font color="Green" title="E68">A8</font></i></a></i>;<br/>
<a NAME="E40:63"/><b>then </b><i><font color="Green" title="E89">A29</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_4.html#T83" title="JGRAPH_4:th.83">JGRAPH_4:83</a></i>;<br/>
<a NAME="E41:63"/><i><font color="Green" title="E90">A30</font></i>: 
( ( ( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E35:63"><i><font color="Green" title="E85">A25</font></i></a>, <a class="txt" href="jgraph_5.html#E37:63"><i><font color="Green" title="E87">A27</font></i></a>, <a class="ref" href="jgraph_5.html#T23" target="_self" title="JGRAPH_5:th.23">Th23</a></i>;<br/>
<a NAME="E42:63"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">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="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E43:63"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E7:63"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T24" target="_self" title="JGRAPH_5:th.24">Th24</a></i>;<br/>
<a NAME="E44:63"/><b>then </b><i><font color="Green" title="E91">A31</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E22:63"><i><font color="Green" title="E72">A12</font></i></a>, <a class="txt" href="jgraph_5.html#E23:63"><i><font color="Green" title="E73">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E24:63"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="jgraph_5.html#E25:63"><i><font color="Green" title="E75">A15</font></i></a>, <a class="txt" href="jgraph_5.html#E36:63"><i><font color="Green" title="E86">A26</font></i></a>, <a class="txt" href="jgraph_5.html#E38:63"><i><font color="Green" title="E88">A28</font></i></a>, <a class="ref" href="jgraph_5.html#T56" target="_self" title="JGRAPH_5:th.56">Th56</a></i>;<br/>
<a NAME="E45:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p3</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="c11">r1</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E16:63"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="jgraph_5.html#E32:63"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E46:63"/><b>then </b><i><font color="Green" title="E92">A32</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T23" target="_self" title="JGRAPH_5:th.23">Th23</a></i>;<br/>
<a NAME="E47:63"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p2</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">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E48:63"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E9:63"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T24" target="_self" title="JGRAPH_5:th.24">Th24</a></i>;<br/>
<a NAME="E49:63"/><b>then </b><i><font color="Green" title="E93">A33</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E24:63"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="jgraph_5.html#E25:63"><i><font color="Green" title="E75">A15</font></i></a>, <a class="txt" href="jgraph_5.html#E26:63"><i><font color="Green" title="E76">A16</font></i></a>, <a class="txt" href="jgraph_5.html#E27:63"><i><font color="Green" title="E77">A17</font></i></a>, <a class="txt" href="jgraph_5.html#E36:63"><i><font color="Green" title="E86">A26</font></i></a>, <a class="txt" href="jgraph_5.html#E46:63"><i><font color="Green" title="E92">A32</font></i></a>, <a class="ref" href="jgraph_5.html#T56" target="_self" title="JGRAPH_5:th.56">Th56</a></i>;<br/>
<a NAME="E50:63"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p3</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="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">p4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="jgraph_5.html#T50" target="_self" title="JGRAPH_5:th.50">Th50</a></i>;<br/>
<a NAME="E51:63"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E5:63"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E11:63"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E18:63"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="jgraph_5.html#T24" target="_self" title="JGRAPH_5:th.24">Th24</a></i>;<br/>
<a NAME="E52:63"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c12">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:63"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E26:63"><i><font color="Green" title="E76">A16</font></i></a>, <a class="txt" href="jgraph_5.html#E27:63"><i><font color="Green" title="E77">A17</font></i></a>, <a class="txt" href="jgraph_5.html#E28:63"><i><font color="Green" title="E78">A18</font></i></a>, <a class="txt" href="jgraph_5.html#E29:63"><i><font color="Green" title="E79">A19</font></i></a>, <a class="txt" href="jgraph_5.html#E40:63"><i><font color="Green" title="E89">A29</font></i></a>, <a class="txt" href="jgraph_5.html#E46:63"><i><font color="Green" title="E92">A32</font></i></a>, <a class="ref" href="jgraph_5.html#T56" target="_self" title="JGRAPH_5:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E53:63"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">q3</font>, <font color="Olive" title="b5">q4</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 <br/>( <font color="Olive" title="b1">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; ( for <font color="Olive" title="b6">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Olive" title="b4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Olive" title="b5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b4">q3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b4">q3</font>,<font color="Olive" title="b5">q4</font>,<font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E20:63"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E21:63"><i><font color="Green" title="E71">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E40:63"><i><font color="Green" title="E89">A29</font></i></a>, <a class="txt" href="jgraph_5.html#E41:63"><i><font color="Green" title="E90">A30</font></i></a>, <a class="txt" href="jgraph_5.html#E44:63"><i><font color="Green" title="E91">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E46:63"><i><font color="Green" title="E92">A32</font></i></a>, <a class="txt" href="jgraph_5.html#E49:63"><i><font color="Green" title="E93">A33</font></i></a></i>;<br/>


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