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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" 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="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="b3">q1</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">q1</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="b4">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp;  not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">C</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="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="c4">q1</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> <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="c2">p1</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> &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;  not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E21">A1</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> &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> )
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c4">q1</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="E3:33"/><span class="lab"><font color="Green" title="E23">A3</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="E4:33"/><span class="lab"><font color="Green" title="E24">A4</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>
 <span class="kw">and </span><br/><a NAME="E5:33"/><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="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
( <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> &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> )
 ; <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>,<font color="Maroon" title="c3">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">P1</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="E8:33"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c6">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="E9:33"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c7">P1</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="E10:33"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">and </span><br/><a NAME="E11:33"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">P1</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="c2">p1</font>,<font color="Maroon" title="c3">p2</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="#E1:33"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="topreal2.html#T5" onmouseover="rs('topreal2/T5')" onmouseout="rh()">TOPREAL2:5</a></span>;<br/>
<a NAME="E12:33"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c6">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="#E10:33"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:33"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <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>  st <font color="Olive" title="b1">A</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> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">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="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/>

<a NAME="E14:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c6">P</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="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="#E8:33"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E29">A9</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="E15:33"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon" title="c7">P1</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="#E10:33"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_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="E1:33_1"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T51" onmouseover="rs('zfmisc_1/T51')" onmouseout="rh()">ZFMISC_1:51</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">and </span><br/><a NAME="E2:33_1_1"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
 not <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</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="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><span class="kw">now </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">  ex <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">A</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="c4">q1</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c8">A</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</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="c8">A</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="c4">q1</font> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><div><span class="lab"><font color="Green" title="E37">A17</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font></span></a><div class="add"><a NAME="E1:33_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</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="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" onmouseover="rs('jordan6/T26')" onmouseout="rh()">JORDAN6:26</a></span>;<br/><span class="kw">assume </span><a NAME="E2:33_1_1_1_1"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</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"> contradiction</span><br/><a NAME="E3:33_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1_1_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="jordan6.html#T55" onmouseover="rs('jordan6/T55')" onmouseout="rh()">JORDAN6:55</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><a NAME="E2:33_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="jordan5c.html#T10" onmouseover="rs('jordan5c/T10')" onmouseout="rh()">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_1"/>
<font color="Maroon" title="c8">A</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="c4">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="jordan16.html#T21" onmouseover="rs('jordan16/T21')" onmouseout="rh()">JORDAN16:21</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"> ( <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E4:33_1_1_1"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T2" onmouseover="rs('jordan16/T2')" onmouseout="rh()">JORDAN16:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:33_1_1_1"/>
<font color="Maroon" title="c8">A</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="#E12:33"><span class="lab"><font color="Green" title="E31">A11</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"> <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><a NAME="E6:33_1_1_1"/>
 not <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E4:33_1_1_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:33_1_1_1"/>
<font color="Maroon" title="c8">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="c3">p2</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="#E1:33_1_1_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T51" onmouseover="rs('zfmisc_1/T51')" onmouseout="rh()">ZFMISC_1:51</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><span class="kw">hence </span><a NAME="E4:33_1_1"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:33_1_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">and </span><br/><a NAME="E2:33_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
 not <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</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="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><span class="kw">now </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">  ex <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">A</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="c4">q1</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c8">A</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</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="c8">A</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="c4">q1</font> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><div><span class="lab"><font color="Green" title="E37">A22</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font></span></a><div class="add"><a NAME="E1:33_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</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="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" onmouseover="rs('jordan6/T26')" onmouseout="rh()">JORDAN6:26</a></span>;<br/><span class="kw">assume </span><a NAME="E2:33_1_2_1_1"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</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"> contradiction</span><br/><a NAME="E3:33_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_2_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="jordan6.html#T55" onmouseover="rs('jordan6/T55')" onmouseout="rh()">JORDAN6:55</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><a NAME="E2:33_1_2_1"/>
<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E36">A21</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="E3:33_1_2_1"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="jordan5c.html#T10" onmouseover="rs('jordan5c/T10')" onmouseout="rh()">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_1_2_1"/>
<font color="Maroon" title="c8">A</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="c4">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="jordan16.html#T21" onmouseover="rs('jordan16/T21')" onmouseout="rh()">JORDAN16:21</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"> ( <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E5:33_1_2_1"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T2" onmouseover="rs('jordan16/T2')" onmouseout="rh()">JORDAN16:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:33_1_2_1"/>
<font color="Maroon" title="c8">A</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="#E15:33"><span class="lab"><font color="Green" title="E34">A14</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"> <font color="Maroon" title="c8">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="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_2"><span class="kw">now </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">  not <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1_2_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">A</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"> contradiction</span><br/><a NAME="E2:33_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E5:33_1_2_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:33_1_2_1"/>
<font color="Maroon" title="c8">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="c3">p2</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="#E1:33_1_2_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T51" onmouseover="rs('zfmisc_1/T51')" onmouseout="rh()">ZFMISC_1:51</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><span class="kw">hence </span><a NAME="E4:33_1_2"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_3"><span class="kw">suppose </span></a><a NAME="E1:33_1_3"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</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="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><a NAME="E2:33_1_3"/>
<font color="Maroon" title="c7">P1</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="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="#E13:33"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E29">A9</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="E3:33_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> or <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T51" onmouseover="rs('zfmisc_1/T51')" onmouseout="rh()">ZFMISC_1:51</a></span>;<br/><a NAME="E4:33_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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> or <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E1:33_1_3"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:33_1_3"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
