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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</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>; <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">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  not <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> holds <br/> <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</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"> (  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> implies  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>) )</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1"/>
( <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> or  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"> (  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> implies  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>) )</span><br/><div class="add"><a NAME="E2:19_1_1"/><span class="kw">then </span>
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T45" onmouseover="rs('jordan1k/T45')" onmouseout="rh()">JORDAN1K:45</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_1"/>
(  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> implies  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1k.html#T44" onmouseover="rs('jordan1k/T44')" onmouseout="rh()">JORDAN1K:44</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="19_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"> (  not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> implies  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>) )</span><br/><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:19_1_2"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 not <font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:19_1_2"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></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"> contradiction</span><br/><a NAME="E4:19_1_2"/><span class="lab"><font color="Green" title="E19">A4</font></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="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">C</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="jordan1k.html#T48" onmouseover="rs('jordan1k/T48')" onmouseout="rh()">JORDAN1K:48</a></span>;<br/><a NAME="E5:19_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="subset_1.html#T29" onmouseover="rs('subset_1/T29')" onmouseout="rh()">SUBSET_1:29</a></span>;<br/><a NAME="E6:19_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T27" onmouseover="rs('jordan2c/T27')" onmouseout="rh()">JORDAN2C:27</a></span>;<br/><a NAME="E7:19_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2"><span class="lab"><font color="Green" title="E17">A2</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">hence </span><a NAME="E8:19_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="jordan11.html#T13" target="_self" onmouseover="rs('jordan11/T13')" onmouseout="rh()">Th13</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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