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

<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font> holds <br/><font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> implies <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> or <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> or <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> or ( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T27" target="_self" onmouseover="rs('jordan18/T27')" onmouseout="rh()">Th27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="32_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_2"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T28" target="_self" onmouseover="rs('jordan18/T28')" onmouseout="rh()">Th28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="32_1_3"><span class="kw">suppose </span></a><a NAME="E1:32_1_3"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_3"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T30" target="_self" onmouseover="rs('jordan18/T30')" onmouseout="rh()">Th30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="32_1_4"><span class="kw">suppose </span></a><a NAME="E1:32_1_4"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_4"/>
<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T29" target="_self" onmouseover="rs('jordan18/T29')" onmouseout="rh()">Th29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="32_1_5"><span class="kw">suppose </span></a><a NAME="E1:32_1_5"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> <a href="jordan18.html#R1" title="JORDAN18:pred.1">-separate</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="jordan18.html#D3" target="_self" onmouseover="rs('jordan18/D3')" onmouseout="rh()">JORDAN18:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E2:32_1_5"/>
( <font color="Maroon" title="c6">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><span class="kw"> such that </span><br/><a NAME="E4:32_1_5"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_5"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E6:32_1_5"/>
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">B</font> <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"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan18.html#T15" target="_self" onmouseover="rs('jordan18/T15')" onmouseout="rh()">Th15</a></span>;<br/><span class="kw">assume </span><a NAME="E7:32_1_5"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E8:32_1_5"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T49" onmouseover="rs('zfmisc_1/T49')" onmouseout="rh()">ZFMISC_1:49</a></span>;<br/><a NAME="E9:32_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E5:32_1_5"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:32_1_5"/>
 not <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_5"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T49" onmouseover="rs('zfmisc_1/T49')" onmouseout="rh()">ZFMISC_1:49</a></span>;<br/><a NAME="E11:32_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:32_1_5"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><span class="kw"> such that </span><br/><a NAME="E13:32_1_5"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E14:32_1_5"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E15:32_1_5"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:32_1_5"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E4:32_1_5"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E9:32_1_5"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="jordan16.html#T23" onmouseover="rs('jordan16/T23')" onmouseout="rh()">JORDAN16:23</a></span>;<br/><a NAME="E16:32_1_5"/>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_5"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E17:32_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_5"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:32_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E13:32_1_5"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E15:32_1_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

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