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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/>

<a NAME="E3:21"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></span>;<br/>
<a NAME="E4:21"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></span>;<br/>
<a NAME="E5:21"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></span>;<br/>
<a NAME="E6:21"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T50" title="JORDAN6:th.50">JORDAN6:50</a></span>;<br/>
<a NAME="E7:21"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T50" title="JORDAN6:th.50">JORDAN6:50</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font> or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> &amp;  not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> ) or ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font> &amp;  not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="jordan16.html#T7" title="JORDAN16:th.7">JORDAN16:7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_1_1"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="jordan_a.html#T16" target="_self" title="JORDAN_A:th.16">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_2"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E4:21_1_2"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="jordan_a.html#T8" target="_self" title="JORDAN_A:th.8">Th9</a></span>;<br/><a NAME="E5:21_1_2"/>
 not <font color="Maroon" title="c3">q</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E3:21_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="jordan_a.html#T5" target="_self" title="JORDAN_A:th.5">Th6</a></span>;<br/><a NAME="E6:21_1_2"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E7:21_1_2"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E4:21_1_2"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_3"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_3"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_3"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E4:21_1_3"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_3_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K9" title="SUBSET_1:func.9">/\</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_1_3_1"/><span class="kw">then </span>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">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#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:21_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="jordan6.html#T60" title="JORDAN6:th.60">JORDAN6:60</a></span>;<br/><a NAME="E4:21_1_3_1"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">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#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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_3_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:21_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E3:21_1_3_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="jordan6.html#T59" title="JORDAN6:th.59">JORDAN6:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:21_1_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_3"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="jordan17.html#T3" title="JORDAN17:th.3">JORDAN17:3</a></span>;<br/><a NAME="E7:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A19</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E8:21_1_3"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan_a.html#T14" target="_self" title="JORDAN_A:th.14">Th15</a></span>;<br/><a NAME="E9:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A21</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_3"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></span>;<br/><a NAME="E10:21_1_3"/>
<font color="Maroon" title="c3">q</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_3"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="jordan17.html#T4" title="JORDAN17:th.4">JORDAN17:4</a></span>;<br/><a NAME="E11:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A22</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E12:21_1_3"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan_a.html#T15" target="_self" title="JORDAN_A:th.15">Th16</a></span>;<br/><a NAME="E13:21_1_3"/><span class="kw">then </span>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_1_3"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></span>;<br/><a NAME="E14:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A24</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K9" title="SUBSET_1:func.9">/\</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_1_3"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E15:21_1_3"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T20" title="JORDAN6:th.20">JORDAN6:20</a></span>;<br/><a NAME="E16:21_1_3"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T19" title="JORDAN6:th.19">JORDAN6:19</a></span>;<br/><a NAME="E17:21_1_3"/>
<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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></span>;<br/><a NAME="E18:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A27</font></span>: 
<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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K9" title="SUBSET_1:func.9">/\</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_3"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E14:21_1_3"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E15:21_1_3"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E16:21_1_3"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="ref" href="topreal8.html#T1" title="TOPREAL8:th.1">TOPREAL8:1</a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E19:21_1_3"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:21_1_3"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E7:21_1_3"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E8:21_1_3"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>;<br/><a NAME="E20:21_1_3"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E4:21_1_3"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E11:21_1_3"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="txt" href="#E12:21_1_3"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>;<br/><a NAME="E21:21_1_3"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_3"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E3:21_1_3"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="jordan_a.html#T12" target="_self" title="JORDAN_A:th.12">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E22:21_1_3"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:21_1_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E19:21_1_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E20:21_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="topreal1.html#T2" title="TOPREAL1:th.2">TOPREAL1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_4"/><span class="lab"><font color="Green" title="E25">A30</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_4"/><span class="lab"><font color="Green" title="E26">A31</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_4"/><span class="lab"><font color="Green" title="E27">A32</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E4:21_1_4"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_4"><span class="lab"><font color="Green" title="E26">A31</font></span></a>, <a class="ref" href="jordan17.html#T3" title="JORDAN17:th.3">JORDAN17:3</a></span>;<br/><a NAME="E5:21_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A33</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E6:21_1_4"/><span class="lab"><font color="Green" title="E29">A34</font></span>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan_a.html#T14" target="_self" title="JORDAN_A:th.14">Th15</a></span>;<br/><a NAME="E7:21_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A35</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_4"><span class="lab"><font color="Green" title="E28">A33</font></span></a>, <a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></span>;<br/><a NAME="E8:21_1_4"/><span class="lab"><font color="Green" title="E31">A36</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E1:21_1_4"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="ref" href="jordan6.html#T21" title="JORDAN6:th.21">JORDAN6:21</a></span>;<br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) = 
<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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_4"><span class="lab"><font color="Green" title="E26">A31</font></span></a>, <a class="txt" href="#E3:21_1_4"><span class="lab"><font color="Green" title="E27">A32</font></span></a>, <a class="ref" href="jordan_a.html#T12" target="_self" title="JORDAN_A:th.12">Th13</a></span>
<br/>.= 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_4"><span class="lab"><font color="Green" title="E30">A35</font></span></a>, <a class="txt" href="#E8:21_1_4"><span class="lab"><font color="Green" title="E31">A36</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:21_1_4"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:21_1_4"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="txt" href="#E5:21_1_4"><span class="lab"><font color="Green" title="E28">A33</font></span></a>, <a class="txt" href="#E6:21_1_4"><span class="lab"><font color="Green" title="E29">A34</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_5"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_5"/><span class="lab"><font color="Green" title="E25">A37</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_5"/><span class="lab"><font color="Green" title="E26">A38</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_5"/><span class="lab"><font color="Green" title="E27">A39</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E4:21_1_5"/>
<font color="Maroon" title="c3">q</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_5"><span class="lab"><font color="Green" title="E27">A39</font></span></a>, <a class="ref" href="jordan17.html#T4" title="JORDAN17:th.4">JORDAN17:4</a></span>;<br/><a NAME="E5:21_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A40</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">q</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E6:21_1_5"/><span class="lab"><font color="Green" title="E29">A41</font></span>: 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan_a.html#T15" target="_self" title="JORDAN_A:th.15">Th16</a></span>;<br/><a NAME="E7:21_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A42</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_5"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></span>;<br/><a NAME="E8:21_1_5"/><span class="lab"><font color="Green" title="E31">A43</font></span>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:21_1_5"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="ref" href="jordan6.html#T23" title="JORDAN6:th.23">JORDAN6:23</a></span>;<br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) = 
<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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_5"><span class="lab"><font color="Green" title="E26">A38</font></span></a>, <a class="txt" href="#E3:21_1_5"><span class="lab"><font color="Green" title="E27">A39</font></span></a>, <a class="ref" href="jordan_a.html#T12" target="_self" title="JORDAN_A:th.12">Th13</a></span>
<br/>.= 
 <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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c3">q</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_5"><span class="lab"><font color="Green" title="E30">A42</font></span></a>, <a class="txt" href="#E8:21_1_5"><span class="lab"><font color="Green" title="E31">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:21_1_5"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E1:21_1_5"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="txt" href="#E5:21_1_5"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="txt" href="#E6:21_1_5"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_6"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_6"/><span class="lab"><font color="Green" title="E25">A44</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_6"/><span class="lab"><font color="Green" title="E26">A45</font></span>: 
 not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E3:21_1_6"/><span class="lab"><font color="Green" title="E27">A46</font></span>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:21_1_6"><span class="lab"><font color="Green" title="E25">A44</font></span></a>, <a class="ref" href="jordan_a.html#T11" target="_self" title="JORDAN_A:th.11">Th12</a></span>;<br/><a NAME="E4:21_1_6"/>
 not <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_6"><span class="lab"><font color="Green" title="E26">A45</font></span></a>, <a class="ref" href="jordan_a.html#T6" target="_self" title="JORDAN_A:th.6">Th7</a></span>;<br/><a NAME="E5:21_1_6"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">C</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_1_6"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c1">C</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E3:21_1_6"><span class="lab"><font color="Green" title="E27">A46</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
