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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>



<b>set </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/>
<b>assume </b><a NAME="E1:104"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">U</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a><br/>

<a NAME="E2:104"/><i><font color="Green">E70</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:104"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:104"><i><font color="Green">E78</font></i></a>, <a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/>
<a NAME="E4:104"/><b>then </b><i><font color="Green">E78</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1"><b>suppose </b></a><a NAME="E1:104_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">TU</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, <a class="txt" href="jordan.html#E1:104_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TU</font> ;<br/><b>reconsider </b><font color="Maroon">f</font> = <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span> ;<br/><b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a><br/><b>thus </b><a NAME="E5:104_1_1"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_2"><b>suppose </b></a><a NAME="E1:104_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:104_1_2"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 <b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E2:104_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/><b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">U</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/><a NAME="E5:104_1_2"/>
<font color="Maroon">V</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E4:104"><i><font color="Green">E78</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/><a NAME="E6:104_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="sprect_3.html#T18">SPRECT_3:18</a></i>;<br/><a NAME="E7:104_1_2"/>
<font color="Maroon">U</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan.html#E4:104"><i><font color="Green">E78</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E8:104_1_2"/><b>then </b>
<font color="Maroon">V</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/><a NAME="E9:104_1_2"/><b>then </b>
<font color="Maroon">V</font> is <a href="topreal4.html#V2">being_Region</a>
 <b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="topreal4.html#D3">TOPREAL4:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:104_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E12:104_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="jordan.html#E6:104_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E1:104_1_2"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="topreal4.html#T30">TOPREAL4:30</a></i>;<br/><a NAME="E13:104_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="topreal4.html#T4">TOPREAL4:4</a></i>;<br/><a NAME="E14:104_1_2"/>
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="topreal4.html#T3">TOPREAL4:3</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><b> such that </b><br/><a NAME="E16:104_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E17:104_1_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E18:104_1_2"/><i><font color="Green">E118</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a><br/><a NAME="E20:104_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E21:104_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, , , <a class="txt" href="jordan.html#E11:104_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>hence </b><a NAME="E22:104_1_2"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><b>thus </b><a NAME="E23:104_1_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
