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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>, <font color="Olive" title="b2">b</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</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">b</font>,<font color="Maroon" title="c1">P</font> holds <br/> ex <font color="Olive" title="b3">c</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="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c1">P</font> implies  ex <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E6">A2</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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">  ex <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/>

<a NAME="E3:8"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_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#K23" title="PSCOMP_1:func.23">W-min</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>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green" title="E8">A4</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="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_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> &amp; <font color="Maroon" title="c3">b</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; not <font color="Maroon" title="c3">b</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#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> &amp; <font color="Maroon" title="c3">b</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> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <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#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">b</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; not <font color="Maroon" title="c3">b</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> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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> ) )
 </span><b>by </b><i><a class="txt" href="#E2:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_1"/><i><font color="Green" title="E9">A5</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>
 <b>and </b><br/><a NAME="E2:8_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c3">b</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>and </b><br/><a NAME="E3:8_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
not <font color="Maroon" title="c3">b</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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><a NAME="E4:8_1_1"/><i><font color="Green" title="E12">A8</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#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>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" title="JORDAN7:th.1">JORDAN7:1</a></i>;<br/><a NAME="E5:8_1_1"/><i><font color="Green" title="E13">A9</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#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/><a NAME="E6:8_1_1"/><i><font color="Green" title="E14">A10</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/><b>thus </b><a NAME="E7:8_1_1"/>
 ex <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</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="8_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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="E1:8_1_1_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; <font color="Maroon" title="c3">b</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> ) or <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="c3">b</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green" title="E15">A11</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>
 <b>and </b><br/><a NAME="E2:8_1_1_1_1_1"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c3">b</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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c4">e</font> =  <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="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E3:8_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="c4">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1_1_1_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="#E2:8_1_1_1_1_1"><i><font color="Green" title="E16">A12</font></i></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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E4:8_1_1_1_1_1"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E3:8_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E4:8_1_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="#E5:8_1_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="#E6:8_1_1"><i><font color="Green" title="E14">A10</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="8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2"/><i><font color="Green" title="E15">A13</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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><a NAME="E2:8_1_1_1_1_2"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="#E4:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">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="E4:8_1_1_1_1_2"/><i><font color="Green" title="E16">A14</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> )
 <b>and </b><br/><a NAME="E5:8_1_1_1_1_2"/><i><font color="Green" title="E17">A15</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">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>and </b><br/><a NAME="E6:8_1_1_1_1_2"/><i><font color="Green" title="E18">A16</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</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="#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E4:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self" title="JORDAN17:th.6">Th6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">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="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E7:8_1_1_1_1_2"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="#E4:8_1_1_1_1_2"><i><font color="Green" title="E16">A14</font></i></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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E8:8_1_1_1_1_2"/><i><font color="Green" title="E19">A17</font></i>: 
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="#E5:8_1_1_1_1_2"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><a NAME="E9:8_1_1_1_1_2"/>
<font color="Maroon" title="c4">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="#E4:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E3:8_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E6:8_1_1_1_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/><b>hence </b><a NAME="E10:8_1_1_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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="#E3:8_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E5:8_1_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="#E1:8_1_1_1_1_2"><i><font color="Green" title="E15">A13</font></i></a>, <a class="txt" href="#E5:8_1_1_1_1_2"><i><font color="Green" title="E17">A15</font></i></a>, <a class="txt" href="#E6:8_1_1_1_1_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="#E8:8_1_1_1_1_2"><i><font color="Green" title="E19">A17</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="8_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_3"/><i><font color="Green" title="E15">A18</font></i>: 
<font color="Maroon" title="c3">b</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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><a NAME="E2:8_1_1_1_1_3"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E3:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="#E1:8_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">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="E4:8_1_1_1_1_3"/><i><font color="Green" title="E16">A19</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> )
 <b>and </b><br/><a NAME="E5:8_1_1_1_1_3"/><i><font color="Green" title="E17">A20</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <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="E6:8_1_1_1_1_3"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E3:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self" title="JORDAN17:th.6">Th6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">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="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E7:8_1_1_1_1_3"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="#E4:8_1_1_1_1_3"><i><font color="Green" title="E16">A19</font></i></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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E8:8_1_1_1_1_3"/>
<font color="Maroon" title="c4">e</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>
 <b>by </b><i><a class="txt" href="#E5:8_1_1_1_1_3"><i><font color="Green" title="E17">A20</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E9:8_1_1_1_1_3"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="#E3:8_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="#E5:8_1_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="#E1:8_1_1_1_1_3"><i><font color="Green" title="E15">A18</font></i></a>, <a class="txt" href="#E5:8_1_1_1_1_3"><i><font color="Green" title="E17">A20</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="8_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_2"/><i><font color="Green" title="E9">A21</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>
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green" title="E10">A22</font></i>: 
<font color="Maroon" title="c3">b</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>
 <b>and </b><br/><a NAME="E3:8_1_2"/><i><font color="Green" title="E11">A23</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c4">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="E5:8_1_2"/><i><font color="Green" title="E12">A24</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> )
 <b>and </b><br/><a NAME="E6:8_1_2"/><i><font color="Green" title="E13">A25</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</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>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E3:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="#E3:8_1_2"><i><font color="Green" title="E11">A23</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self" title="JORDAN17:th.6">Th6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">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="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E7:8_1_2"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="#E5:8_1_2"><i><font color="Green" title="E12">A24</font></i></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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E8:8_1_2"/>
<font color="Maroon" title="c4">e</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>
 <b>by </b><i><a class="txt" href="#E6:8_1_2"><i><font color="Green" title="E13">A25</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E9:8_1_2"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_2"><i><font color="Green" title="E9">A21</font></i></a>, <a class="txt" href="#E2:8_1_2"><i><font color="Green" title="E10">A22</font></i></a>, <a class="txt" href="#E6:8_1_2"><i><font color="Green" title="E13">A25</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="8_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_3"/><i><font color="Green" title="E9">A26</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>and </b><br/><a NAME="E2:8_1_3"/><i><font color="Green" title="E10">A27</font></i>: 
<font color="Maroon" title="c3">b</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>and </b><br/><a NAME="E3:8_1_3"/><i><font color="Green" title="E11">A28</font></i>: 
not <font color="Maroon" title="c3">b</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>
 <b>and </b><br/><a NAME="E4:8_1_3"/><i><font color="Green" title="E12">A29</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>
 ; <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">c</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">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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">c</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">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c4">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="E6:8_1_3"/><i><font color="Green" title="E13">A30</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font> )
 <b>and </b><br/><a NAME="E7:8_1_3"/><i><font color="Green" title="E14">A31</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">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>and </b><br/><a NAME="E8:8_1_3"/><i><font color="Green" title="E15">A32</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</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="#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E4:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E4:8_1_3"><i><font color="Green" title="E12">A29</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self" title="JORDAN17:th.6">Th6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">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="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><b>thus </b><a NAME="E9:8_1_3"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="#E6:8_1_3"><i><font color="Green" title="E13">A30</font></i></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="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E10:8_1_3"/><i><font color="Green" title="E16">A33</font></i>: 
<font color="Maroon" title="c4">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="#E4:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="#E3:8_1_3"><i><font color="Green" title="E11">A28</font></i></a>, <a class="txt" href="#E8:8_1_3"><i><font color="Green" title="E15">A32</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/><a NAME="E11:8_1_3"/>
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="#E7:8_1_3"><i><font color="Green" title="E14">A31</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E12:8_1_3"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="#E1:8_1_3"><i><font color="Green" title="E9">A26</font></i></a>, <a class="txt" href="#E2:8_1_3"><i><font color="Green" title="E10">A27</font></i></a>, <a class="txt" href="#E3:8_1_3"><i><font color="Green" title="E11">A28</font></i></a>, <a class="txt" href="#E7:8_1_3"><i><font color="Green" title="E14">A31</font></i></a>, <a class="txt" href="#E8:8_1_3"><i><font color="Green" title="E15">A32</font></i></a>, <a class="txt" href="#E10:8_1_3"><i><font color="Green" title="E16">A33</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>
