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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c1">C</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><b>let </b><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> implies  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/>

<a NAME="E2:23"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> ) or ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_1"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="jordan_a.html#T20" target="_self" title="JORDAN_A:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/>
( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><div class="add"><a NAME="E2:23_1_2"/><b>then </b>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="jordan7.html#T8" title="JORDAN7:th.8">JORDAN7:8</a></i>;<br/><b>hence </b><a NAME="E3:23_1_2"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_3"><b>suppose </b></a><a NAME="E1:23_1_3"/>
( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_3"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="jordan_a.html#T18" target="_self" title="JORDAN_A:th.18">Th18</a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
