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<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V2" title="COMPTS_1:attr.2">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>; <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 <font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> &amp;  <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">P</font> holds <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> &amp; <font color="Olive" title="b2">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> )</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"> ( <font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> &amp;  <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">P</font> implies ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> ) )</span><br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> &amp;  <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">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font></span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
( <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">P</font> or <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">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
<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">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font></span><br/><div class="add"><a NAME="E2:10_1_1_1"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan7.html#T5" target="_self" title="JORDAN7:th.5">Th5</a></i>;<br/><a NAME="E3:10_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/><a NAME="E4:10_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</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="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
<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">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font></span><br/><div class="add"><a NAME="E2:10_1_1_2"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan7.html#T5" target="_self" title="JORDAN7:th.5">Th5</a></i>;<br/><a NAME="E3:10_1_1_2"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/><a NAME="E4:10_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p1</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">P</font> ) ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:10_1_1_2"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font>
 <b>by </b><i><a class="txt" href="#E1:10_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</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>
<b>end;</b></div>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2"/>
( <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">P</font> or <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">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1"><b>suppose </b></a><a NAME="E1:10_2_1"/><i><font color="Green" title="E11">A4</font></i>: 
<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">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font></span><br/><div class="add"><a NAME="E2:10_2_1"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan7.html#T5" target="_self" title="JORDAN7:th.5">Th5</a></i>;<br/><a NAME="E3:10_2_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/><a NAME="E4:10_2_1"/><b>then </b>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:10_2_1"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font>
 <b>by </b><i><a class="txt" href="#E1:10_2_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</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="10_2_2"><b>suppose </b></a><a NAME="E1:10_2_2"/><i><font color="Green" title="E11">A5</font></i>: 
<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">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font></span><br/><div class="add"><a NAME="E2:10_2_2"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p1</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">P</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_2_2"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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">P</font>
 <b>by </b><i><a class="txt" href="#E1:10_2_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</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>
