<?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="E1:68"/><i><font color="Green" title="E62">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> )
 ;<br/>

<a NAME="E2:68"/><b>then </b><i><font color="Green" title="E63">A2</font></i>: 
<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="E3:68"/><b>then </b><i><font color="Green" title="E64">A3</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green" title="E65">A4</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> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green" title="E63">A2</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">p44</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="E6:68"/><i><font color="Green" title="E66">A5</font></i>: 
( <font color="Maroon" title="c6">p44</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">p44</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/>
<a NAME="E7:68"/><i><font color="Green" title="E67">A6</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#E6:68"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="jgraph_5.html#T3" target="_self" title="JGRAPH_5:th.3">Th3</a></i>;<br/>
<a NAME="E8:68"/><i><font color="Green" title="E68">A7</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <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">p7</font> where <font color="Olive" title="b1">p7</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> : ( <font color="Olive" title="b1">p7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p7</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T38" target="_self" title="JGRAPH_5:th.38">Th38</a></i>;<br/>
<a NAME="E9:68"/><i><font color="Green" title="E69">A8</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <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">p7</font> where <font color="Olive" title="b1">p7</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> : ( <font color="Olive" title="b1">p7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p7</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self" title="JGRAPH_5:th.37">Th37</a></i>;<br/>
<a NAME="E10:68"/><i><font color="Green" title="E70">A9</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>;<br/>
<a NAME="E11:68"/><b>then </b><i><font color="Green" title="E71">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_1"/><i><font color="Green" title="E72">A11</font></i>: 
<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>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add">1 = 
<span class="p1">(<span class="default"><span class="p2">(<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="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="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#E6:68"><i><font color="Green" title="E66">A5</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/>.= 
<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> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68_1_1_1"><i><font color="Green" title="E72">A11</font></i></a></i>
;<br/><a NAME="E3:68_1_1_1"/><b>then </b><i><font color="Green" title="E73">A12</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:68_1_1_1"/><b>then </b><i><font color="Green" title="E74">A13</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:68"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/><a NAME="E5:68_1_1_1"/><i><font color="Green" title="E75">A14</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E10:68"><i><font color="Green" title="E70">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><div><i><font color="Green" title="E76">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1_2_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> or  not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:68_1_1_1_2_1_1"/><i><font color="Green" title="E77">A16</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_2_1_1"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:68"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E5:68_1_1_1"><i><font color="Green" title="E75">A14</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:68_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:68_1_1_1"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_1_2_1_1"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:68_1_1_1_2_1_2"/>
not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:68"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/><a NAME="E3:68_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:68"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E8:68"><i><font color="Green" title="E68">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:68_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E11:68"><i><font color="Green" title="E71">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_1"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68_1_1_1_2_1_2"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:68_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E8:68_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E9:68_1_1_1"/><b>then </b><i><font color="Green" title="E77">A18</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_1"><i><font color="Green" title="E76">A15</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><a NAME="E10:68_1_1_1"/><i><font color="Green" title="E78">A19</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_1"><i><font color="Green" title="E76">A15</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E11:68_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E12:68_1_1_1"/><b>then </b><i><font color="Green" title="E79">A20</font></i>: 
<font color="Maroon" title="c2">p2</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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E10:68_1_1_1"><i><font color="Green" title="E78">A19</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><a NAME="E13:68_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E10:68_1_1_1"><i><font color="Green" title="E78">A19</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E14:68_1_1_1"/><b>then </b>
<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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E15:68_1_1_1"/>
 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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68_1_1_1"><i><font color="Green" title="E77">A18</font></i></a>, <a class="txt" href="jgraph_5.html#E12:68_1_1_1"><i><font color="Green" title="E79">A20</font></i></a>, <a class="ref" href="jgraph_5.html#T62" target="_self" title="JGRAPH_5:th.62">Th62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2"/><i><font color="Green" title="E72">A21</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_1_2"/><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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:68"><i><font color="Green" title="E67">A6</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="c7">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="E4:68_1_1_2"/><i><font color="Green" title="E73">A22</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="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <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>  )
 <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="c8">r1</font> = <font color="Maroon" title="c7">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="E6:68_1_1_2"/><i><font color="Green" title="E74">A23</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="c8">r1</font> &amp; <font color="Maroon" title="c8">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#E7:68"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2"><i><font color="Green" title="E73">A22</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="c9">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="E8:68_1_1_2"/><i><font color="Green" title="E75">A24</font></i>: 
( <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">r1</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a>  &amp; <font color="Maroon" title="c9">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#T143" title="JGRAPH_4:th.143">JGRAPH_4:143</a></i>;<br/><b>set </b><font color="Maroon" title="c10">q11</font> = <font color="Maroon" title="c9">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="c11">q22</font> = <font color="Maroon" title="c9">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="c12">q33</font> = <font color="Maroon" title="c9">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="c13">q44</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_1_1"/>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">p4</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_1"/><i><font color="Green" title="E76">A25</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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">p4</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:68_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E77">A26</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T52" target="_self" title="JGRAPH_5:th.52">Th52</a></i>;<br/><a NAME="E3:68_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E78">A27</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T52" target="_self" title="JGRAPH_5:th.52">Th52</a></i>;<br/><a NAME="E4:68_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E79">A28</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T52" target="_self" title="JGRAPH_5:th.52">Th52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1_1_2_1_1_1_1"/><i><font color="Green" title="E80">A29</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="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#E6:68"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></i>
<br/>.= 
<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="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68_1_1_2_1_1_1_1"><i><font color="Green" title="E80">A29</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:68_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68_1_1_2"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_2_1_1_1_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="square_1.html#T109" title="SQUARE_1:th.109">SQUARE_1:109</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:68_1_1_2_1_1_1"/>
 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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_2_1_1_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68_1_1_2_1_1_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_2_1_1_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2_1_1_1"><i><font color="Green" title="E79">A28</font></i></a>, <a class="ref" href="jgraph_5.html#T66" target="_self" title="JGRAPH_5:th.66">Th66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2"/><i><font color="Green" title="E76">A30</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:68_1_1_2_1_1_2"/>
<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#K10" title="REAL_1:func.10">/</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#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">r1</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E6:68"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2"><i><font color="Green" title="E73">A22</font></i></a></i>;<br/><a NAME="E3:68_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E77">A31</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:68_1_1_2"><i><font color="Green" title="E73">A22</font></i></a>, <a class="txt" href="jgraph_5.html#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="txt" href="jgraph_5.html#E1:68_1_1_2_1_1_2"><i><font color="Green" title="E76">A30</font></i></a>, <a class="ref" href="jgraph_5.html#T29" target="_self" title="JGRAPH_5:th.29">Th29</a></i>;<br/><a NAME="E4:68_1_1_2_1_1_2"/><i><font color="Green" title="E78">A32</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">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="c9">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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_2"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="jgraph_5.html#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="jgraph_5.html#T61" target="_self" title="JGRAPH_5:th.61">Th61</a></i>;<br/><a NAME="E5:68_1_1_2_1_1_2"/><i><font color="Green" title="E79">A33</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">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="c9">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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_2"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="jgraph_5.html#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="jgraph_5.html#T61" target="_self" title="JGRAPH_5:th.61">Th61</a></i>;<br/><a NAME="E6:68_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E80">A34</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">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="c9">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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2_1_1_2"><i><font color="Green" title="E78">A32</font></i></a>, <a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E7:68_1_1_2_1_1_2"/><i><font color="Green" title="E81">A35</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">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="c9">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#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_2"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="jgraph_5.html#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="jgraph_5.html#T61" target="_self" title="JGRAPH_5:th.61">Th61</a></i>;<br/><a NAME="E8:68_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>;<br/><a NAME="E9:68_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E82">A36</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><div><i><font color="Green" title="E83">A37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_1_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:68_1_1_2_1_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_2_1_1_2"><i><font color="Green" title="E77">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2_1_1_2"><i><font color="Green" title="E78">A32</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68_1_1_2_1_1_2"><i><font color="Green" title="E82">A36</font></i></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" title="JGRAPH_5:th.51">Th51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green" title="E84">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_1_2_1_1_2_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:68_1_1_2_1_1_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_2_1_1_2"><i><font color="Green" title="E77">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2_1_1_2"><i><font color="Green" title="E78">A32</font></i></a>, <a class="txt" href="jgraph_5.html#E5:68_1_1_2_1_1_2"><i><font color="Green" title="E79">A33</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68_1_1_2_1_1_2"><i><font color="Green" title="E82">A36</font></i></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" title="JGRAPH_5:th.51">Th51</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_1_1_2_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_1_2_1_1_2_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_1_1_2_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:68_1_1_2_1_1_2_3_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_2_1_1_2"><i><font color="Green" title="E77">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E6:68_1_1_2_1_1_2"><i><font color="Green" title="E80">A34</font></i></a>, <a class="txt" href="jgraph_5.html#E7:68_1_1_2_1_1_2"><i><font color="Green" title="E81">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68_1_1_2_1_1_2"><i><font color="Green" title="E82">A36</font></i></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" title="JGRAPH_5:th.51">Th51</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">f2</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>, <font color="Maroon" title="c15">q81</font>, <font color="Maroon" title="c16">q82</font>, <font color="Maroon" title="c17">q83</font>, <font color="Maroon" title="c18">q84</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="E14:68_1_1_2_1_1_2"/><i><font color="Green" title="E85">A39</font></i>: 
( <font color="Maroon" title="c14">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; ( 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="c14">f2</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> ) &amp; <font color="Maroon" title="c15">q81</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> &amp; <font color="Maroon" title="c16">q82</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> &amp; <font color="Maroon" title="c17">q83</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> &amp; <font color="Maroon" title="c18">q84</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> &amp; <font color="Maroon" title="c15">q81</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c15">q81</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c16">q82</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c16">q82</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">q83</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">q83</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">q84</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">q84</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c15">q81</font>,<font color="Maroon" title="c16">q82</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c16">q82</font>,<font color="Maroon" title="c17">q83</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c17">q83</font>,<font color="Maroon" title="c18">q84</font>,<font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68_1_1_2_1_1_2"><i><font color="Green" title="E77">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E4:68_1_1_2_1_1_2"><i><font color="Green" title="E78">A32</font></i></a>, <a class="txt" href="jgraph_5.html#E5:68_1_1_2_1_1_2"><i><font color="Green" title="E79">A33</font></i></a>, <a class="txt" href="jgraph_5.html#E7:68_1_1_2_1_1_2"><i><font color="Green" title="E81">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E10:68_1_1_2_1_1_2"><i><font color="Green" title="E83">A37</font></i></a>, <a class="txt" href="jgraph_5.html#E11:68_1_1_2_1_1_2"><i><font color="Green" title="E84">A38</font></i></a>, <a class="ref" href="jgraph_5.html#T66" target="_self" title="JGRAPH_5:th.66">Th66</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">f3</font> = <font color="Maroon" title="c14">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font> as   <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> ;<br/><a NAME="E16:68_1_1_2_1_1_2"/><i><font color="Green" title="E86">A40</font></i>: 
<font color="Maroon" title="c19">f3</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="ref" href="tops_2.html#T71" title="TOPS_2:th.71">TOPS_2:71</a></i>;<br/><a NAME="E17:68_1_1_2_1_1_2"/><i><font color="Green" title="E87">A41</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="c19">f3</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">q</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/>

<a NAME="E1:68_1_1_2_1_1_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:68_1_1_2_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><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="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<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="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">q</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c20">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#E8:68_1_1_2"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="jgraph_4.html#T135" title="JGRAPH_4:th.135">JGRAPH_4:135</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E18:68_1_1_2_1_1_2"/><i><font color="Green" title="E88">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:68_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E89">A43</font></i>: 
<font color="Maroon" title="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q81</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E20:68_1_1_2_1_1_2"/><i><font color="Green" title="E90">A44</font></i>: 
<font color="Maroon" title="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q82</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E18:68_1_1_2_1_1_2"><i><font color="Green" title="E88">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E21:68_1_1_2_1_1_2"/><i><font color="Green" title="E91">A45</font></i>: 
<font color="Maroon" title="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q83</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E18:68_1_1_2_1_1_2"><i><font color="Green" title="E88">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E22:68_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q84</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E18:68_1_1_2_1_1_2"><i><font color="Green" title="E88">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E23:68_1_1_2_1_1_2"/>
 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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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#E14:68_1_1_2_1_1_2"><i><font color="Green" title="E85">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E16:68_1_1_2_1_1_2"><i><font color="Green" title="E86">A40</font></i></a>, <a class="txt" href="jgraph_5.html#E17:68_1_1_2_1_1_2"><i><font color="Green" title="E87">A41</font></i></a>, <a class="txt" href="jgraph_5.html#E19:68_1_1_2_1_1_2"><i><font color="Green" title="E89">A43</font></i></a>, <a class="txt" href="jgraph_5.html#E20:68_1_1_2_1_1_2"><i><font color="Green" title="E90">A44</font></i></a>, <a class="txt" href="jgraph_5.html#E21:68_1_1_2_1_1_2"><i><font color="Green" title="E91">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:68_1_1_2"/>
 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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:68"/>
 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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &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> )
 ;<br/>


</div>
