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<span class="kw">let </span><font color="Maroon" title="c1">P</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>; <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">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" 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="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> 
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">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">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E3:64"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan6.html#D9" target="_self" onmouseover="rs('jordan6/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E4:64"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="jordan6.html#T50" target="_self" onmouseover="rs('jordan6/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_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"> ( ( <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI: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="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  not <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI: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="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_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="E1:64_1_1"/>
( <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  not <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E44">A3</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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><span class="kw">case </span></a><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">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">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:64_1_1_1"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</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#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5c.html#T9" onmouseover="rs('jordan5c/T9')" onmouseout="rh()">JORDAN5C:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2"><span class="kw">case </span></a><a NAME="E1:64_1_1_2"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
( <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  not <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">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">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:64_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5c.html#T9" onmouseover="rs('jordan5c/T9')" onmouseout="rh()">JORDAN5C:9</a></span>;<br/><a NAME="E3:64_1_1_2"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E1:64_1_1_2"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="topreal1.html#T1" onmouseover="rs('topreal1/T1')" onmouseout="rh()">TOPREAL1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E47">A6</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></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:64"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">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"> verum</span><br/>


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