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

<b>let </b><font color="Maroon" title="c1">D</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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>assume </b><a NAME="E1:67"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E2:67"/>
<font color="Maroon" title="c1">D</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73" title="JORDAN2C:th.73">JORDAN2C:73</a></i>;<br/>
<a NAME="E3:67"/><b>then </b>
<font color="Maroon" title="c2">C</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="jordan2c.html#T16" title="JORDAN2C:th.16">JORDAN2C:16</a></i>;<br/>
<a NAME="E4:67"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T88" title="TOPREAL6:th.88">TOPREAL6:88</a></i>;<br/>
<b>assume </b><a NAME="E5:67"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E6:67"/><i><font color="Green" title="E57">A4</font></i>: 
for <font color="Olive" title="b1">C</font> being <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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>  st not <font color="Olive" title="b1">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</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> ex <font color="Olive" title="b4">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> st <br/>( <font color="Olive" title="b4">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">C</font> be  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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>assume </b><a NAME="E1:67_1"/>
not <font color="Maroon" title="c3">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</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><b> such that </b><br/><a NAME="E3:67_1"/><i><font color="Green" title="E58">A5</font></i>: 
( <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d2</font> )
 <b>by </b><i><a class="ref" href="realset1.html#T15" title="REALSET1:th.15">REALSET1:15</a></i>;<br/>
<b>assume </b><a NAME="E4:67_1"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<b>assume </b><a NAME="E5:67_1"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E6:67_1"/><b>then </b>
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E4:67_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">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><b> such that </b><br/><a NAME="E8:67_1"/><i><font color="Green" title="E60">A7</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="borsuk_4.html#T1" target="_self" title="BORSUK_4:th.1">Th1</a></i>;<br/>
<a NAME="E9:67_1"/><i><font color="Green" title="E61">A8</font></i>: 
( <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Dp</font> = <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> 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="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a></i>;<br/>
<a NAME="E11:67_1"/>
<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="c7">Dp</font>, <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="borsuk_4.html#T77" target="_self" title="BORSUK_4:th.77">Th77</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">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> <font color="Maroon" title="c7">Dp</font></span>)</span>,<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <b> such that </b><br/><a NAME="E13:67_1"/><i><font color="Green" title="E62">A9</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></i>;<br/>
<a NAME="E14:67_1"/><i><font color="Green" title="E63">A10</font></i>: 
<font color="Maroon" title="c3">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> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">C'</font> = <font color="Maroon" title="c3">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> <font color="Maroon" title="c7">Dp</font></span>)</span> <b>by </b><i><a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">fC</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">C'</font>;<br/>
<a NAME="E16:67_1"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E8:67_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E17:67_1"/><b>then </b><i><font color="Green" title="E64">A11</font></i>: 
<font color="Maroon" title="c9">C'</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#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E18:67_1"/><i><font color="Green" title="E65">A12</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_1"><i><font color="Green" title="E62">A9</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="E19:67_1"/><i><font color="Green" title="E66">A13</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">fC</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">C'</font> as  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E17:67_1"><i><font color="Green" title="E64">A11</font></i></a>, <a class="txt" href="borsuk_4.html#E18:67_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="compts_1.html#T24" title="COMPTS_1:th.24">COMPTS_1:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">fC'</font> = <font color="Maroon" title="c11">fC</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E22:67_1"/><i><font color="Green" title="E67">A14</font></i>: 
<font color="Maroon" title="c12">fC'</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 <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> 
 
;<br/>
<a NAME="E23:67_1"/><i><font color="Green" title="E68">A15</font></i>: 
<font color="Maroon" title="c12">fC'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> 
 
;<br/>
<a NAME="E24:67_1"/><i><font color="Green" title="E69">A16</font></i>: 
<font color="Maroon" title="c12">fC'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E22:67_1"><i><font color="Green" title="E67">A14</font></i></a>, <a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></i>;<br/>
<a NAME="E25:67_1"/><i><font color="Green" title="E70">A17</font></i>: 
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>   st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fC'</font> holds <br/><font color="Olive" title="b1">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
;<br/>
<a NAME="E26:67_1"/><i><font color="Green" title="E71">A18</font></i>: 
<font color="Maroon" title="c4">d1</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> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:67_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E27:67_1"/><i><font color="Green" title="E72">A19</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">C'</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E28:67_1"/><i><font color="Green" title="E73">A20</font></i>: 
<font color="Maroon" title="c5">d2</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> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:67_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E29:67_1"/><i><font color="Green" title="E74">A21</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">C'</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">fC'</font> = <font color="Maroon" title="c12">fC'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E23:67_1"><i><font color="Green" title="E68">A15</font></i></a>, <a class="txt" href="borsuk_4.html#E25:67_1"><i><font color="Green" title="E70">A17</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<a NAME="E31:67_1"/><i><font color="Green" title="E75">A22</font></i>: 
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E26:67_1"><i><font color="Green" title="E71">A18</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E32:67_1"/><i><font color="Green" title="E76">A23</font></i>: 
<font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E28:67_1"><i><font color="Green" title="E73">A20</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">p1</font>, <font color="Maroon" title="c15">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E34:67_1"/><i><font color="Green" title="E77">A24</font></i>: 
( <font color="Maroon" title="c14">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p2</font> &amp; <font color="Maroon" title="c13">fC'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E24:67_1"><i><font color="Green" title="E69">A16</font></i></a>, <a class="ref" href="borsuk_4.html#T83" target="_self" title="BORSUK_4:th.83">Th83</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">E</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="borsuk_4.html#T49" target="_self" title="BORSUK_4:th.49">Th49</a></i>;<br/>
<a NAME="E36:67_1"/><i><font color="Green" title="E78">A25</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_1"><i><font color="Green" title="E62">A9</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="E37:67_1"/>
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">p2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_1"/>
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p2</font>
 ;<br/>

<a NAME="E2:67_1_1"/><b>then </b><i><font color="Green" title="E79">A26</font></i>: 
<font color="Maroon" title="c13">fC'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c14">p1</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></i>;<br/>
<a NAME="E3:67_1_1"/><b>then </b><i><font color="Green" title="E80">A27</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E27:67_1"><i><font color="Green" title="E72">A19</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:67_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E29:67_1"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="borsuk_4.html#E2:67_1_1"><i><font color="Green" title="E79">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:67_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="borsuk_4.html#E31:67_1"><i><font color="Green" title="E75">A22</font></i></a>, <a class="txt" href="borsuk_4.html#E32:67_1"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="borsuk_4.html#E36:67_1"><i><font color="Green" title="E78">A25</font></i></a>, <a class="txt" href="borsuk_4.html#E3:67_1_1"><i><font color="Green" title="E80">A27</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:67_1"/><b>then </b><i><font color="Green" title="E79">A28</font></i>: 
<font color="Maroon" title="c14">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">p2</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E39:67_1"/><i><font color="Green" title="E80">A29</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E19:67_1"><i><font color="Green" title="E66">A13</font></i></a>, <a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></i>;<br/>
<a NAME="E40:67_1"/><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c16">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E19:67_1"><i><font color="Green" title="E66">A13</font></i></a>, <a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></i>;<br/>
<a NAME="E41:67_1"/><i><font color="Green" title="E82">A31</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/>
<a NAME="E42:67_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E43:67_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">B</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">E</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="borsuk_4.html#E40:67_1"><i><font color="Green" title="E81">A30</font></i></a>, <a class="txt" href="borsuk_4.html#E41:67_1"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="relat_1.html#T174" title="RELAT_1:th.174">RELAT_1:174</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E45:67_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E46:67_1"/><b>then </b><i><font color="Green" title="E83">A32</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Dp</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E47:67_1"/><i><font color="Green" title="E84">A33</font></i>: 
<font color="Maroon" title="c7">Dp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E48:67_1"/>
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">E</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="borsuk_4.html#E39:67_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">s1</font>, <font color="Maroon" title="c19">s2</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><b> such that </b><br/><a NAME="E50:67_1"/><i><font color="Green" title="E85">A34</font></i>: 
<font color="Maroon" title="c17">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="borsuk_4.html#E38:67_1"><i><font color="Green" title="E79">A28</font></i></a>, <a class="txt" href="borsuk_4.html#E41:67_1"><i><font color="Green" title="E82">A31</font></i></a>, <a class="txt" href="borsuk_4.html#E46:67_1"><i><font color="Green" title="E83">A32</font></i></a>, <a class="ref" href="borsuk_4.html#T80" target="_self" title="BORSUK_4:th.80">Th80</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c18">s1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c19">s2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c17">B</font>
;<br/>

<b>thus </b><a NAME="E51:67_1"/>
<font color="Maroon" title="c17">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E50:67_1"><i><font color="Green" title="E85">A34</font></i></a></i>;<br/>

<a NAME="E52:67_1"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E14:67_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E53:67_1"/>
( <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E34:67_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="borsuk_4.html#E41:67_1"><i><font color="Green" title="E82">A31</font></i></a>, <a class="txt" href="borsuk_4.html#E46:67_1"><i><font color="Green" title="E83">A32</font></i></a>, <a class="txt" href="borsuk_4.html#E47:67_1"><i><font color="Green" title="E84">A33</font></i></a>, <a class="ref" href="funct_1.html#T163" title="FUNCT_1:th.163">FUNCT_1:163</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2"/>
( <font color="Maroon" title="c2">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> or  not <font color="Maroon" title="c2">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1"><b>suppose </b></a><a NAME="E1:67_2_1"/><i><font color="Green" title="E58">A35</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2_1_1"/>
( <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">C</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_2_1_1_1"/><i><font color="Green" title="E59">A36</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">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><b> such that </b><br/><a NAME="E3:67_2_1_1_1"/><i><font color="Green" title="E60">A37</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="ref" href="topreal2.html#T4" title="TOPREAL2:th.4">TOPREAL2:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">CC</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <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="E5:67_2_1_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">CC</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">CC</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:67_2_1_1_1"/><b>then </b><i><font color="Green" title="E61">A38</font></i>: 
not <font color="Maroon" title="c5">CC</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_2_1_1_1"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="realset1.html#T14" title="REALSET1:th.14">REALSET1:14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">pp</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c7">qq</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <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="E8:67_2_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pp</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">qq</font>
 <b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/><a NAME="E9:67_2_1_1_1"/><b>then </b><i><font color="Green" title="E62">A39</font></i>: 
<font color="Maroon" title="c5">CC</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#T19" title="COMPTS_1:th.19">COMPTS_1:19</a></i>;<br/><a NAME="E10:67_2_1_1_1"/><i><font color="Green" title="E63">A40</font></i>: 
<font color="Maroon" title="c5">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_2_1_1_1"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E11:67_2_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_2_1_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E2:67_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E64">A41</font></i>: 
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">E</font> = <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">CC</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> <font color="Maroon" title="c1">D</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:67_2_1_1_1_1"/>
not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</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">D</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_2_1_1_1"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="borsuk_4.html#E2:67_2_1_1_1_1"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="jordan6.html#T60" title="JORDAN6:th.60">JORDAN6:60</a></i>;<br/>
<b>hence </b><a NAME="E5:67_2_1_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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>, <font color="Maroon" title="c10">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="E13:67_2_1_1_1"/><i><font color="Green" title="E64">A42</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c5">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:67"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="borsuk_4.html#E6:67_2_1_1_1"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="borsuk_4.html#E9:67_2_1_1_1"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="borsuk_4.html#E10:67_2_1_1_1"><i><font color="Green" title="E63">A40</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">p1</font>
;<br/><b>take </b>
<font color="Maroon" title="c9">p2</font>
;<br/><b>take </b>
<font color="Maroon" title="c10">B</font>
;<br/><b>thus </b><a NAME="E14:67_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_2_1_1_1"><i><font color="Green" title="E64">A42</font></i></a></i>;<br/><b>thus </b><a NAME="E15:67_2_1_1_1"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:67_2_1_1_1"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>thus </b><a NAME="E16:67_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E13:67_2_1_1_1"><i><font color="Green" title="E64">A42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_2_1_1_2"/>
<font color="Maroon" title="c2">C</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/><div class="add"><b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="E3:67_2_1_1_2"/><i><font color="Green" title="E59">A43</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:67_2_1"><i><font color="Green" title="E58">A35</font></i></a>, <a class="ref" href="borsuk_4.html#T2" target="_self" title="BORSUK_4:th.2">Th2</a></i>;<br/><a NAME="E4:67_2_1_1_2"/><i><font color="Green" title="E60">A44</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_2_1_1_2"><i><font color="Green" title="E59">A43</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E6:67_2_1_1_2"/><i><font color="Green" title="E61">A45</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="borsuk_4.html#T3" target="_self" title="BORSUK_4:th.3">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>reconsider </b><font color="Maroon" title="c6">Y</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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>reconsider </b><font color="Maroon" title="c7">Cy</font> = <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="borsuk_4.html#E4:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="compts_1.html#T19" title="COMPTS_1:th.19">COMPTS_1:19</a></i>;<br/><a NAME="E10:67_2_1_1_2"/><i><font color="Green" title="E62">A46</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Cy</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E11:67_2_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:67_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Cy</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:67_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">A47</font></i>: 
not <font color="Maroon" title="c7">Cy</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E4:67_2_1_1_2"><i><font color="Green" title="E60">A44</font></i></a>, <a class="txt" href="borsuk_4.html#E6:67_2_1_1_2"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="borsuk_4.html#E10:67_2_1_1_2"><i><font color="Green" title="E62">A46</font></i></a>, <a class="ref" href="realset1.html#T14" title="REALSET1:th.14">REALSET1:14</a></i>;<br/><a NAME="E14:67_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 ;<br/><a NAME="E15:67_2_1_1_2"/><b>then </b><i><font color="Green" title="E64">A48</font></i>: 
<font color="Maroon" title="c7">Cy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E16:67_2_1_1_2"/>
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_2_1_1_2_1"/>
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E2:67_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E65">A49</font></i>: 
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">E</font> = <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">Cy</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> <font color="Maroon" title="c1">D</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:67_2_1_1_2_1"/>
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:67_2_1_1_2"><i><font color="Green" title="E59">A43</font></i></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/>
<a NAME="E5:67_2_1_1_2_1"/><b>then </b>
not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</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">D</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="borsuk_4.html#E4:67_2_1_1_2"><i><font color="Green" title="E60">A44</font></i></a>, <a class="txt" href="borsuk_4.html#E6:67_2_1_1_2"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="borsuk_4.html#E2:67_2_1_1_2_1"><i><font color="Green" title="E65">A49</font></i></a>, <a class="ref" href="jordan6.html#T60" title="JORDAN6:th.60">JORDAN6:60</a></i>;<br/>
<b>hence </b><a NAME="E6:67_2_1_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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>, <font color="Maroon" title="c10">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="E18:67_2_1_1_2"/><i><font color="Green" title="E65">A50</font></i>: 
( <font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c7">Cy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:67"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="borsuk_4.html#E13:67_2_1_1_2"><i><font color="Green" title="E63">A47</font></i></a>, <a class="txt" href="borsuk_4.html#E15:67_2_1_1_2"><i><font color="Green" title="E64">A48</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">p1</font>
;<br/><b>take </b>
<font color="Maroon" title="c9">p2</font>
;<br/><b>take </b>
<font color="Maroon" title="c10">B</font>
;<br/><b>thus </b><a NAME="E19:67_2_1_1_2"/>
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E18:67_2_1_1_2"><i><font color="Green" title="E65">A50</font></i></a></i>;<br/><b>thus </b><a NAME="E20:67_2_1_1_2"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E10:67_2_1_1_2"><i><font color="Green" title="E62">A46</font></i></a>, <a class="txt" href="borsuk_4.html#E18:67_2_1_1_2"><i><font color="Green" title="E65">A50</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>thus </b><a NAME="E21:67_2_1_1_2"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E18:67_2_1_1_2"><i><font color="Green" title="E65">A50</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_2"><b>suppose </b></a><a NAME="E1:67_2_2"/>
not <font color="Maroon" title="c2">C</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_2_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b3">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="borsuk_4.html#E4:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="borsuk_4.html#E5:67"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E6:67"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
