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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E2:13"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="jordan14.html#T9" target="_self" title="JORDAN14:th.9">Th9</a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="jordan14.html#T10" target="_self" title="JORDAN14:th.10">Th10</a></i>;<br/>
<b>assume </b><a NAME="E4:13"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:13"/>
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">L</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:13"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E8:13"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E9:13"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">R</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:13"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E12:13"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E13:13"/>
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:13_1"/><b>then </b>
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> &amp; not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E3:13_1"/><b>then </b>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">C1</font> = <font color="Maroon" title="c1">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E15:13"/>
<font color="Maroon" title="c6">C1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<b>hence </b><a NAME="E16:13"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan14.html#E3:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="jordan14.html#E4:13"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="jordan14.html#E7:13"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="jordan14.html#E8:13"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="jordan14.html#E11:13"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="jordan14.html#E12:13"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="jordan2c.html#T100" title="JORDAN2C:th.100">JORDAN2C:100</a>, <a class="ref" href="sprect_4.html#T7" title="SPRECT_4:th.7">SPRECT_4:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
