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<b>let </b><font color="Maroon" title="c1">P</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">a</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="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/> ex <font color="Olive" title="b2">e</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</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="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> implies  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> ) )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/>

<a NAME="E2:7"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</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>
 
<b>by </b><i><a class="ref" href="topreal5.html#T25" title="TOPREAL5:th.25">TOPREAL5:25</a></i>;<br/>
<a NAME="E3:7"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( <font color="Maroon" title="c2">a</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> or <font color="Maroon" title="c2">a</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c2">a</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">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>take </b>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E2:7_1_1"/>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</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="ref" href="sprect_1.html#T16" title="SPRECT_1:th.16">SPRECT_1:16</a></i>;<br/><b>hence </b><a NAME="E3:7_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:7_1_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="jordan7.html#T3" title="JORDAN7:th.3">JORDAN7:3</a>, <a class="ref" href="topreal5.html#T25" title="TOPREAL5:th.25">TOPREAL5:25</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="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c2">a</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">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>thus </b><a NAME="E2:7_1_2"/>
 ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_2_1_1"/>
( <font color="Maroon" title="c2">a</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="c1">P</font> or  not <font color="Maroon" title="c2">a</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="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_1_1_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c2">a</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="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>thus </b><a NAME="E2:7_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</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">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c3">e</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="E3:7_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E10">A8</font></i>: 
( <font color="Maroon" title="c3">e</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="c1">P</font> &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">e</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> )
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="jordan6.html#T55" title="JORDAN6:th.55">JORDAN6:55</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">e</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E4:7_1_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:7_1_2_1_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="#E1:7_1_2_1_1_1_1_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="#E3:7_1_2_1_1_1_1_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</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="7_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</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">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>take </b>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E2:7_1_2_1_1_1_1_1_2"/>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</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="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" title="JORDAN7:th.1">JORDAN7:1</a></i>;<br/><b>hence </b><a NAME="E3:7_1_2_1_1_1_1_1_2"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="#E1:7_1_2_1_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="#E1:7_1_2_1_1_1_1_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_1_1_2"/><i><font color="Green" title="E8">A10</font></i>: 
not <font color="Maroon" title="c2">a</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="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</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="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><a NAME="E2:7_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E3:7_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E9">A11</font></i>: 
<font color="Maroon" title="c2">a</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="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E1:7_1_2_1_1_2"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span>, <font color="Maroon" title="c4">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:7_1_2_1_1_2"/><i><font color="Green" title="E10">A12</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>and </b><br/><a NAME="E6:7_1_2_1_1_2"/><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E7:7_1_2_1_1_2"/><i><font color="Green" title="E12">A14</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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>
 <b>and </b><br/><a NAME="E8:7_1_2_1_1_2"/><i><font color="Green" title="E13">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>and </b><br/><a NAME="E9:7_1_2_1_1_2"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E10:7_1_2_1_1_2"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="jordan17.html#T1" target="_self" title="JORDAN17:th.1">Th1</a></i>;<br/><i><font color="Green" title="E16">A18</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> 
<b>by </b><i><a class="txt" href="#E5:7_1_2_1_1_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>
;<br/><a NAME="E12:7_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E8:7_1_2_1_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="#E9:7_1_2_1_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E13:7_1_2_1_1_2"/><i><font color="Green" title="E18">A20</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E11:7_1_2_1_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E14:7_1_2_1_1_2"/><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="#E5:7_1_2_1_1_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E15:7_1_2_1_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="#E1:7_1_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="#E7:7_1_2_1_1_2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="#E9:7_1_2_1_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="#E10:7_1_2_1_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">s</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="E17:7_1_2_1_1_2"/><i><font color="Green" title="E20">A22</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/><a NAME="E18:7_1_2_1_1_2"/><i><font color="Green" title="E21">A23</font></i>: 
<font color="Maroon" title="c5">s</font> is   <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="E19:7_1_2_1_1_2"/><i><font color="Green" title="E22">A24</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="#E8:7_1_2_1_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="#E17:7_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:7_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E11:7_1_2_1_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><i><font color="Green" title="E24">A26</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:7_1_2_1_1_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span>

<br/>.= 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
;<br/><a NAME="E22:7_1_2_1_1_2"/><i><font color="Green" title="E25">A27</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E20:7_1_2_1_1_2"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">e</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> as   <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>by </b><i><a class="txt" href="#E21:7_1_2_1_1_2"><i><font color="Green" title="E24">A26</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">e</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E24:7_1_2_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="#E10:7_1_2_1_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="#E12:7_1_2_1_1_2"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="#E14:7_1_2_1_1_2"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="#E17:7_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="#E20:7_1_2_1_1_2"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <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">a</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font></span><br/><a NAME="E25:7_1_2_1_1_2"/><i><font color="Green" title="E26">A28</font></i>: 
<font color="Maroon" title="c6">e</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>
 <b>by </b><i><a class="txt" href="#E7:7_1_2_1_1_2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="#E13:7_1_2_1_1_2"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="#E14:7_1_2_1_1_2"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="#E17:7_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="#E20:7_1_2_1_1_2"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E26:7_1_2_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="#E5:7_1_2_1_1_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="#E6:7_1_2_1_1_2"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="#E7:7_1_2_1_1_2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="#E8:7_1_2_1_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="#E9:7_1_2_1_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="#E10:7_1_2_1_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="#E17:7_1_2_1_1_2"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="#E18:7_1_2_1_1_2"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="#E19:7_1_2_1_1_2"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="jordan5c.html#T8" title="JORDAN5C:th.8">JORDAN5C:8</a></i>;<br/><b>hence </b><a NAME="E27:7_1_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E3:7_1_2_1_1_2"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="#E21:7_1_2_1_1_2"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="#E22:7_1_2_1_1_2"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="#E25:7_1_2_1_1_2"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</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><b>end;</b></div></div><b>end;</b></div>

</div>
