<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">q2</font>, <font color="Maroon" title="c4">q3</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>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font> &amp; (  not <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font> or  not <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> ) &amp; <font color="Maroon" title="c2">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q3</font> &amp; (  not <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q3</font> or  not <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<b>thus </b><a NAME="E3:14"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font></span>)</span>
 ;<br/>

<a NAME="E2:14_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1"/>
( <font color="Maroon" title="c4">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1"><b>case </b></a><a NAME="E1:14_1_1_1_1"/>
<font color="Maroon" title="c4">q3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan7.html#E2:14_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E4:14_1_1_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_1_1"/>
( <font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan7.html#E2:14_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/><a NAME="E3:14_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">p2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> )
 ;<br/><b>hence </b><a NAME="E4:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan7.html#E4:14_1_1_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_2"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan7.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan7.html#E1:14_1_1_1_1_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2"><b>case </b></a><a NAME="E1:14_1_1_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan7.html#E2:14_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E4:14_1_1_1_2"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> ) ) )
 ;<br/><a NAME="E5:14_1_1_1_2"/>
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan7.html#E2:14_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan7.html#E1:14_1_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="jordan7.html#E4:14_1_1_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/><a NAME="E6:14_1_1_1_2"/><b>then </b>
 ex <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 <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font> )
 ;<br/><b>hence </b><a NAME="E7:14_1_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan7.html#E1:14_1_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="jordan7.html#E4:14_1_1_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:14"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E5:14"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:14"/><b>then </b><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="jordan7.html#T6" target="_self" title="JORDAN7:th.6">Th6</a></i>;<br/>
<a NAME="E7:14"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan7.html#E5:14"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="jordan7.html#T6" target="_self" title="JORDAN7:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E8:14"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c4">q3</font>,<font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:14"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
