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<b>let </b><font color="Maroon" title="c1">Kb</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>;<br/>

<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">Kb</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="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c2">v</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<a NAME="E2:35"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E3:35"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">Kbd</font> = <font color="Maroon" title="c1">Kb</font> as  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> <b>by </b><i><a class="txt" href="jgraph_3.html#E1:35"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>
<a NAME="E5:35"/>
 <a href="topgrp_1.html#K4" title="TOPGRP_1:func.4">id</a> <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> <font color="Maroon" title="c3">Kbd</font></span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T20" title="TOPGRP_1:th.20">TOPGRP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c1">Kb</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:35"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="jgraph_3.html#T34" target="_self" title="JGRAPH_3:th.34">Th34</a></i>;<br/>

<b>set </b><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c1">Kb</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>,<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> <font color="Maroon" title="c1">Kb</font></span>)</span><b> such that </b><br/><a NAME="E8:35"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="jgraph_3.html#E6:35"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="topreal2.html#D1" title="TOPREAL2:def.1">TOPREAL2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon" title="c1">Kb</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c1">Kb</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c1">Kb</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:35_1_1"/>
<font color="Maroon" title="c3">Kbd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:35_1_1"/>
<font color="Maroon" title="c1">Kb</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:35_1_1"><i><font color="Green" title="E48">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/>
not <font color="Maroon" title="c1">Kb</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">R</font> = <font color="Maroon" title="c1">Kb</font> as  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> ;<br/><a NAME="E3:35_1_2"/>
( <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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> <font color="Maroon" title="c1">Kb</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E8:35"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E4:35_1_2"/><b>then </b>
<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> <font color="Maroon" title="c6">R</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/><b>hence </b><a NAME="E5:35_1_2"/>
<font color="Maroon" title="c1">Kb</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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