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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Olive" title="b1">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#K3" title="JORDAN6:func.3">L_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> implies  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">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#K3" title="JORDAN6:func.3">L_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> )</span><br/>



<b>set </b><font color="Maroon" title="c3">q</font> =  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">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#K3" title="JORDAN6:func.3">L_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:16"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">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">C</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="jordan17.html#T3" title="JORDAN17:th.3">JORDAN17:3</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</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">C</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" title="JORDAN7:th.1">JORDAN7:1</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</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">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>means </b>(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,$1,<font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; $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">C</font> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>means </b> <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,$1, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <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>means </b> <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> $1, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <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>means </b>( <font color="Maroon">S<sub>2</sub></font>[$1] or <font color="Maroon">S<sub>3</sub></font>[$1] );<br/>
<a NAME="E5:16"/><i><font color="Green" title="E17">A5</font></i>: 
for <font color="Olive" title="b1">p1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">p1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p1</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p1</font>] )</span><br/>

<b>thus </b><a NAME="E1:16_1"/>
( ( not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c1">C</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">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">C</font> ) ) or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p1</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>] )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">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">C</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<div><i><font color="Green" title="E19">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1_1"/>
( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</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">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">C</font> or <font color="Maroon" title="c4">p1</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">C</font> )</span><br/><a NAME="E2:16_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:16_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1"/>
( <font color="Maroon" title="c4">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">C</font> or <font color="Maroon" title="c4">p1</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">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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="16_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_2"/>
( <font color="Maroon" title="c4">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">C</font> or <font color="Maroon" title="c4">p1</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">C</font> or ( not <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> ) or ( not <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> ) )
 </span><b>by </b><i><a class="txt" href="#E2:16_1_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2_1"><b>suppose </b></a><a NAME="E1:16_1_1_2_1"/>
<font color="Maroon" title="c4">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">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_2_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="jordan5c.html#T9" title="JORDAN5C:th.9">JORDAN5C:9</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="16_1_1_2_2"><b>suppose </b></a><a NAME="E1:16_1_1_2_2"/>
<font color="Maroon" title="c4">p1</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">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_2_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C: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="16_1_1_2_3"><b>suppose </b></a><a NAME="E1:16_1_1_2_3"/>
( not <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_2_3"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E18">A6</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="16_1_1_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:16_1_1_2_4"/><i><font color="Green" title="E20">A8</font></i>: 
not <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:16_1_1_2_4"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:16_1_1_2_4"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E2:16_1_1_2_4"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/><a NAME="E4:16_1_1_2_4"/>
<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</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#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E5:16_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_1_2_4"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="#E3:16_1_1_2_4"><i><font color="Green" title="E22">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>hence </b><a NAME="E6:16_1_1_2_4"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C: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>

<b>assume </b><a NAME="E2:16_1"/><i><font color="Green" title="E18">A11</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]</span><br/>

<a NAME="E3:16_1"/><i><font color="Green" title="E19">A12</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 </span><b>by </b><i><a class="txt" href="#E2:16_1"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1"><b>suppose </b></a><a NAME="E1:16_1_2_1"/><i><font color="Green" title="E20">A13</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]</span><br/><div class="add"><a NAME="E2:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:16_1_2_1"><i><font color="Green" title="E20">A13</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="16_1_2_2"><b>suppose </b></a><a NAME="E1:16_1_2_2"/><i><font color="Green" title="E20">A14</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">p1</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]</span><br/><div class="add"><a NAME="E2:16_1_2_2"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="#E1:16_1_2_2"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_2_1_1"/>
( <font color="Maroon" title="c4">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">C</font> or <font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">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">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:16_1_2_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:16_1"><i><font color="Green" title="E19">A12</font></i></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="16_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_2_1_1_2"/>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:16_1_2_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E2:16_1_2_2"><i><font color="Green" title="E21">A15</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><b>hence </b><a NAME="E4:16_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p1</font>]
 ; <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>
<b>set </b><font color="Maroon" title="c4">Y1</font> = <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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">Y2</font> = <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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/>
<b>set </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">Y'</font> = <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> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p1</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p1</font>] ) </span>}</span> ;<br/>
<a NAME="E6:16"/><i><font color="Green" title="E18">A16</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="#E5:16"><i><font color="Green" title="E17">A5</font></i></a>);<br/></i>
<a NAME="E7:16"/><i><font color="Green" title="E19">A17</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan7.html#D1" title="JORDAN7:def.1">JORDAN7:def 1</a></i>;<br/>
<a NAME="E8:16"/><i><font color="Green" title="E20">A18</font></i>: 
 <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#D3" title="JORDAN6:def.3">JORDAN6:def 3</a></i>;<br/>
<a NAME="E9:16"/>
<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> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p1</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p1</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p1</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="topreal1.html#S1" title="TOPREAL1:sch.1">TOPREAL1:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E10:16"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">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#K3" title="JORDAN6:func.3">L_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:16"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="#E7:16"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E20">A18</font></i></a>, <a class="ref" href="jordan6.html#D4" title="JORDAN6:def.4">JORDAN6:def 4</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>
