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<b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">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>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font>
 <b>and </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan11.html#E3:18_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/><a NAME="E6:18_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="jordan11.html#E4:18_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="jordan1k.html#T50" title="JORDAN1K:th.50">JORDAN1K:50</a></i>;<br/><a NAME="E7:18_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan1a.html#T15" title="JORDAN1A:th.15">JORDAN1A:15</a></i>;<br/><a NAME="E8:18_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="jordan11.html#E2:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="jordan11.html#E4:18_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E9:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan11.html#E3:18"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="jordan11.html#E3:18_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="jordan11.html#E6:18_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="jordan1k.html#T30" title="JORDAN1K:th.30">JORDAN1K:30</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:18"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E19">A8</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T132" title="JORDAN2C:th.132">JORDAN2C:132</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E20">A9</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E1:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E21">A10</font></i>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D5" title="JORDAN2C:def.5">JORDAN2C:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:18"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E12:18"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan11.html#E2:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">B</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><b> such that </b><br/><a NAME="E14:18"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E15:18"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan11.html#E12:18"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/>
<a NAME="E16:18"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan11.html#E15:18"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a></i>;<br/>
<a NAME="E17:18"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E18:18"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">q</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E11:18"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="jordan11.html#E14:18"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E19:18"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E5:18"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="jordan11.html#E6:18"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan11.html#E8:18"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="jordan11.html#E16:18"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="jordan9.html#T3" title="JORDAN9:th.3">JORDAN9:3</a></i>;<br/>
<a NAME="E20:18"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E9:18"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="jordan11.html#E12:18"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="jordan11.html#E14:18"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E21:18"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" title="JORDAN2C:th.28">JORDAN2C:28</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E22:18"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan11.html#E19:18"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="xboole_1.html#T66" title="XBOOLE_1:th.66">XBOOLE_1:66</a></i>;<br/>


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