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<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  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 <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="E1:60"/><i><font color="Green" title="E51">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/>

<b>assume </b><a NAME="E2:60"/><i><font color="Green" title="E52">A2</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
( <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="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/>
<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="E2:60_1_1"/>
(  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> st <font color="Maroon" title="c2">C</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> or  ex <font color="Olive" title="b1">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> st <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="Olive" title="b1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_4.html#T2" target="_self" title="BORSUK_4:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_1_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>then consider </b><font color="Maroon" title="c3">d1</font>, <font color="Maroon" title="c4">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="E3:60_1_2"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c4">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c3">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d2</font> )
 <b>by </b><i><a class="ref" href="realset1.html#T15" title="REALSET1:th.15">REALSET1:15</a></i>;<br/><a NAME="E4:60_1_2"/>
<font color="Maroon" title="c2">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#E1:60"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="borsuk_4.html#E2:60"><i><font color="Green" title="E52">A2</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="c5">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="E6:60_1_2"/><i><font color="Green" title="E54">A4</font></i>: 
( <font color="Maroon" title="c5">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="c2">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="c5">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_4.html#T1" target="_self" title="BORSUK_4:th.1">Th1</a></i>;<br/><a NAME="E7:60_1_2"/><i><font color="Green" title="E55">A5</font></i>: 
( <font color="Maroon" title="c3">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="c5">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">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="c5">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#E3:60_1_2"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E6:60_1_2"><i><font color="Green" title="E54">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">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="c5">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#E3:60_1_2"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E6:60_1_2"><i><font color="Green" title="E54">A4</font></i></a></i>;<br/><a NAME="E9:60_1_2"/>
<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">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#E6:60_1_2"><i><font color="Green" title="E54">A4</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="c7">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="c6">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="E11:60_1_2"/><i><font color="Green" title="E56">A6</font></i>: 
<font color="Maroon" title="c7">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/><b>reconsider </b><font color="Maroon" title="c8">C'</font> = <font color="Maroon" title="c2">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="c6">Dp</font></span>)</span> <b>by </b><i><a class="txt" href="borsuk_4.html#E6:60_1_2"><i><font color="Green" title="E54">A4</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="c9">fC</font> = <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font>;<br/><a NAME="E13:60_1_2"/>
<font color="Maroon" title="c2">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="c6">Dp</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:60_1_2"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E14:60_1_2"/><b>then </b><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c8">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="E15:60_1_2"/><i><font color="Green" title="E58">A8</font></i>: 
<font color="Maroon" title="c8">C'</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/><a NAME="E16:60_1_2"/><i><font color="Green" title="E59">A9</font></i>: 
<font color="Maroon" title="c7">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#E11:60_1_2"><i><font color="Green" title="E56">A6</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="E17:60_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">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#E11:60_1_2"><i><font color="Green" title="E56">A6</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="c10">fC</font> = <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font> as  <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="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E14:60_1_2"><i><font color="Green" title="E57">A7</font></i></a>, <a class="txt" href="borsuk_4.html#E15:60_1_2"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="borsuk_4.html#E16:60_1_2"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="compts_1.html#T24" title="COMPTS_1:th.24">COMPTS_1:24</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">fC'</font> = <font color="Maroon" title="c10">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="E20:60_1_2"/><i><font color="Green" title="E60">A10</font></i>: 
<font color="Maroon" title="c11">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="E21:60_1_2"/><i><font color="Green" title="E61">A11</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="c11">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="E22:60_1_2"/><i><font color="Green" title="E62">A12</font></i>: 
<font color="Maroon" title="c3">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="c6">Dp</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E7:60_1_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E23:60_1_2"/><i><font color="Green" title="E63">A13</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:60_1_2"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E24:60_1_2"/><i><font color="Green" title="E64">A14</font></i>: 
<font color="Maroon" title="c4">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="c6">Dp</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E7:60_1_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E25:60_1_2"/><i><font color="Green" title="E65">A15</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:60_1_2"><i><font color="Green" title="E53">A3</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="c12">fC'</font> = <font color="Maroon" title="c11">fC'</font> 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#E20:60_1_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="borsuk_4.html#E21:60_1_2"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="compts_1.html#T11" title="COMPTS_1:th.11">COMPTS_1:11</a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/><a NAME="E27:60_1_2"/><i><font color="Green" title="E66">A16</font></i>: 
<font color="Maroon" title="c3">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="c7">f</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E22:60_1_2"><i><font color="Green" title="E62">A12</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="E28:60_1_2"/><i><font color="Green" title="E67">A17</font></i>: 
<font color="Maroon" title="c4">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="c7">f</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E24:60_1_2"><i><font color="Green" title="E64">A14</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="c13">p1</font>, <font color="Maroon" title="c14">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="E30:60_1_2"/><i><font color="Green" title="E68">A18</font></i>: 
( <font color="Maroon" title="c13">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">p2</font> &amp; <font color="Maroon" title="c12">fC'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_4.html#T56" target="_self" title="BORSUK_4:th.56">Th56</a></i>;<br/><a NAME="E31:60_1_2"/><i><font color="Green" title="E69">A19</font></i>: 
<font color="Maroon" title="c7">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#E11:60_1_2"><i><font color="Green" title="E56">A6</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="E32:60_1_2"/>
<font color="Maroon" title="c13">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1_2_1"/>
<font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p2</font>
 ;<br/>

<a NAME="E2:60_1_2_1"/><b>then </b><i><font color="Green" title="E70">A20</font></i>: 
<font color="Maroon" title="c12">fC'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c13">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#E30:60_1_2"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></i>;<br/>
<a NAME="E3:60_1_2_1"/><b>then </b><i><font color="Green" title="E71">A21</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p1</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E23:60_1_2"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:60_1_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p1</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E25:60_1_2"><i><font color="Green" title="E65">A15</font></i></a>, <a class="txt" href="borsuk_4.html#E2:60_1_2_1"><i><font color="Green" title="E70">A20</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="E5:60_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:60_1_2"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E27:60_1_2"><i><font color="Green" title="E66">A16</font></i></a>, <a class="txt" href="borsuk_4.html#E28:60_1_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="borsuk_4.html#E31:60_1_2"><i><font color="Green" title="E69">A19</font></i></a>, <a class="txt" href="borsuk_4.html#E3:60_1_2_1"><i><font color="Green" title="E71">A21</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="E33:60_1_2"/><b>then </b>
<font color="Maroon" title="c13">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">p2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E30:60_1_2"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">s1</font>, <font color="Maroon" title="c16">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="E35:60_1_2"/><i><font color="Green" title="E70">A22</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c15">s1</font>,<font color="Maroon" title="c16">s2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:60_1_2"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="borsuk_4.html#E11:60_1_2"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="borsuk_4.html#E30:60_1_2"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="borsuk_4.html#T80" target="_self" title="BORSUK_4:th.80">Th80</a></i>;<br/><b>thus </b><a NAME="E36:60_1_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> st <font color="Maroon" title="c2">C</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> or  ex <font color="Olive" title="b1">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> st <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="Olive" title="b1">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#E35:60_1_2"><i><font color="Green" title="E70">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
