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<b>set </b><font color="Maroon" title="c7">S</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #);<br/>
<b>set </b><font color="Maroon" title="c8">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #);<br/>
<a NAME="E1:20_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T14" target="_self" title="TOPALG_3:th.14">Th14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  not <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
not <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c9">J</font> =  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/><b>given </b><font color="Maroon" title="c10">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span><b> such that </b><a NAME="E2:20_1_1_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E3:20_1_1_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/><a NAME="E4:20_1_1_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T8" title="PCOMPS_1:th.8">PCOMPS_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">j0</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c12">j1</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:20_1_1_1_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">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"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></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="E7:20_1_1_1_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E8:20_1_1_1_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">xy0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:20_1_1_1_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c13">xy0</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="c10">f</font>
 <b>and </b><br/><a NAME="E11:20_1_1_1_1_1"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j0</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">x0</font>, <font color="Maroon" title="c15">y0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:20_1_1_1_1_1"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c14">x0</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #)
 <b>and </b><br/><a NAME="E14:20_1_1_1_1_1"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c15">y0</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)
 <b>and </b><br/><a NAME="E15:20_1_1_1_1_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c13">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x0</font>,<font color="Maroon" title="c15">y0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E7:20_1_1_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topalg_3.html#E10:20_1_1_1_1_1"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">xy1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:20_1_1_1_1_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c16">xy1</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="c10">f</font>
 <b>and </b><br/><a NAME="E18:20_1_1_1_1_1"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">xy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j1</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E8:20_1_1_1_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">x1</font>, <font color="Maroon" title="c18">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:20_1_1_1_1_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c17">x1</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #)
 <b>and </b><br/><a NAME="E21:20_1_1_1_1_1"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c18">y1</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)
 <b>and </b><br/><a NAME="E22:20_1_1_1_1_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c16">xy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x1</font>,<font color="Maroon" title="c18">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E7:20_1_1_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topalg_3.html#E17:20_1_1_1_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x0</font> = <font color="Maroon" title="c14">x0</font>, <font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c17">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #) <b>by </b><i><a class="txt" href="topalg_3.html#E13:20_1_1_1_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topalg_3.html#E20:20_1_1_1_1_1"><i><font color="Green" title="E24">A16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">y0</font> = <font color="Maroon" title="c15">y0</font>, <font color="Maroon" title="c22">y1</font> = <font color="Maroon" title="c18">y1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #) <b>by </b><i><a class="txt" href="topalg_3.html#E14:20_1_1_1_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="topalg_3.html#E21:20_1_1_1_1_1"><i><font color="Green" title="E25">A17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">S</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #), <font color="Maroon" title="c24">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="borsuk_3.html#T4" title="BORSUK_3:th.4">BORSUK_3:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">X0</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c19">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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 <font color="Maroon" title="c23">S</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">Y1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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 <font color="Maroon" title="c24">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E28:20_1_1_1_1_1"/>
<font color="Maroon" title="c24">T</font>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="borsuk_3.html#T15" title="BORSUK_3:th.15">BORSUK_3:15</a></i>;<br/><a NAME="E29:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T25" title="TOPREAL6:th.25">TOPREAL6:25</a></i>;<br/><a NAME="E30:20_1_1_1_1_1"/>
<font color="Maroon" title="c23">S</font>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span>,<font color="Maroon" title="c23">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="borsuk_3.html#T15" title="BORSUK_3:th.15">BORSUK_3:15</a></i>;<br/><a NAME="E31:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E28">A20</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span>,<font color="Maroon" title="c23">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T25" title="TOPREAL6:th.25">TOPREAL6:25</a></i>;<br/><a NAME="E32:20_1_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span>,<font color="Maroon" title="c23">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="yellow12.html#T44" title="YELLOW12:th.44">YELLOW12:44</a></i>;<br/><a NAME="E33:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E31:20_1_1_1_1_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="topreal6.html#T25" title="TOPREAL6:th.25">TOPREAL6:25</a></i>;<br/><b>set </b><font color="Maroon" title="c27">g</font> = <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>;<br/><a NAME="E34:20_1_1_1_1_1"/><i><font color="Green" title="E30">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E35:20_1_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c24">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E36:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <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"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E37:20_1_1_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c28">g</font> = <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="topalg_3.html#E36:20_1_1_1_1_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> = 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">X0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">X0</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c24">T</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>
;<br/><a NAME="E40:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E32">A24</font></i>: 
<font color="Maroon" title="c28">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><b>set </b><font color="Maroon" title="c29">h</font> = <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>;<br/><a NAME="E41:20_1_1_1_1_1"/><i><font color="Green" title="E33">A25</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E42:20_1_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c23">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E43:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E34">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <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"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E44:20_1_1_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> <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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c30">h</font> = <font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="topalg_3.html#E43:20_1_1_1_1_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> = 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">Y1</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">S</font>,<font color="Maroon" title="c24">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c23">S</font></span>)</span>,<font color="Maroon" title="c26">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>
;<br/><a NAME="E47:20_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E35">A27</font></i>: 
<font color="Maroon" title="c30">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><div><i><font color="Green" title="E36">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1_1_1_1_3"/><i><font color="Green" title="E37">A29</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:20_1_1_1_1_1_3"/>
<font color="Maroon" title="c28">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_1_1_1_3_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c28">g</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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c31">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:20_1_1_1_1_1_3_1"/><i><font color="Green" title="E38">A30</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ;<br/>

<a NAME="E3:20_1_1_1_1_1_3_1"/><i><font color="Green" title="E39">A31</font></i>: 
<font color="Maroon" title="c19">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1_1_1_3_1_1"/>
( <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_3_1"><i><font color="Green" title="E38">A30</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E40">A32</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E41">A33</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c21">y0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c28">g</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c28">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c21">y0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">y</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E11:20_1_1_1_1_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="topalg_3.html#E15:20_1_1_1_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E40">A32</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c28">g</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E41">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E40">A34</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c10">f</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topalg_3.html#E7:20_1_1_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topalg_3.html#E13:20_1_1_1_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topalg_3.html#E21:20_1_1_1_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E41">A35</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:20_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E42">A36</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c28">g</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon" title="c28">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c31">y</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E40">A34</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E41">A35</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:20_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c28">g</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E29:20_1_1_1_1_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self" title="TOPALG_3:th.8">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1_1_1_1_4"/><i><font color="Green" title="E37">A37</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:20_1_1_1_1_1_4"/>
<font color="Maroon" title="c30">h</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_1_1_1_4_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c30">h</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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c31">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:20_1_1_1_1_1_4_1"/><i><font color="Green" title="E38">A38</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 ;<br/>

<a NAME="E3:20_1_1_1_1_1_4_1"/><i><font color="Green" title="E39">A39</font></i>: 
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Y1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1_1_1_4_1_1"/>
( <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_4_1"><i><font color="Green" title="E38">A38</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_4_1_1_1"/><i><font color="Green" title="E40">A40</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_4_1_1_1"/><i><font color="Green" title="E41">A41</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c30">h</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">y</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E18:20_1_1_1_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="topalg_3.html#E22:20_1_1_1_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4_1_1_1"><i><font color="Green" title="E40">A40</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c30">h</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_4_1_1_1"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_4_1_1_2"/><i><font color="Green" title="E40">A42</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_4_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c10">f</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="topalg_3.html#E7:20_1_1_1_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="topalg_3.html#E13:20_1_1_1_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="topalg_3.html#E21:20_1_1_1_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E41">A43</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:20_1_1_1_1_1_4_1_1_2"/><i><font color="Green" title="E42">A44</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c30">h</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon" title="c30">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x0</font>,<font color="Maroon" title="c22">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c31">y</font>
<b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4"><i><font color="Green" title="E37">A37</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4_1_1_2"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1_1_2"><i><font color="Green" title="E41">A43</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:20_1_1_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c30">h</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1_4_1_1_2"><i><font color="Green" title="E42">A44</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="topalg_3.html#E47:20_1_1_1_1_1"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self" title="TOPALG_3:th.8">Th8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E50:20_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E48:20_1_1_1_1_1"><i><font color="Green" title="E36">A28</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self" title="TOPALG_3:th.8">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green" title="E10">A45</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">A</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">B</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">B</font> #)</span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_2"/>
for <font color="Olive" title="b1">T</font> being <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>  holds  <font color="Olive" title="b1">T</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E3:20_1_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_2"><i><font color="Green" title="E10">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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