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

<b>set </b><font color="Maroon" title="c1">GG</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">SS</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>] <b>means </b>$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>means </b>$1 <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">$1 <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">K</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">L</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<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="E4:27"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c5">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T24" target="_self" title="BORSUK_6:th.24">Th24</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="topreal6.html#T85" title="TOPREAL6:th.85">TOPREAL6:85</a></i>;<br/>
<a NAME="E6:27"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:27_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">L</font>
  <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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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="E1:27_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c7">y</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="c5">f</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E5:27_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">z</font>] )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:27_1_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c8">z</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"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
;<br/>
<a NAME="E7:27_1_1"/><b>then </b>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </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/>
<b>then consider </b><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:27_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c9">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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon" title="c10">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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  &amp; <font color="Maroon" title="c8">z</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="c9">x1</font>,<font color="Maroon" title="c10">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><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="c11">x1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c10">y1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="borsuk_6.html#E9:27_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:27_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">y1</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E3:27_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="borsuk_6.html#E5:27_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="borsuk_6.html#E9:27_1_1"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">x'</font> = <font color="Maroon" title="c6">x</font> as   <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> ;<br/>
<a NAME="E13:27_1_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:27_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E14:27_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:27_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E15:27_1_1"/>
( <font color="Maroon" title="c8">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> &amp; <font color="Maroon" title="c8">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y1</font> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E9:27_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E16:27_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E5:27_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="borsuk_6.html#E13:27_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="borsuk_6.html#E14:27_1_1"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:27_1"/>
<font color="Maroon" title="c4">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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="E1:27_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">z</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="E3:27_1_2"/><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">z</font>] )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:27_1_2"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
<a NAME="E6:27_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:27_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E7:27_1_2"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:27_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E5:27_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">Y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E8:27_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:27_1_2"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">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> 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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_6.html#E8:27_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:27_1_2"/>
( <font color="Maroon" title="c11">Y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E12:27_1_2"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:27_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>
<a NAME="E13:27_1_2"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c11">Y</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="c5">f</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E9:27_1_2"><i><font color="Green" title="E28">A13</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="E14:27_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="borsuk_6.html#E3:27_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E5:27_1_2"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:27_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E12:27_1_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="borsuk_6.html#E13:27_1_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> : <font color="Olive" title="b1">p</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span> </span>}</span>  is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E5:27"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="borsuk_6.html#T21" target="_self" title="BORSUK_6:th.21">Th21</a>, <a class="ref" href="tops_2.html#T72" title="TOPS_2:th.72">TOPS_2:72</a></i>;<br/>


</div>
