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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="borsuk_4.html#E1:47"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">E</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="borsuk_4.html#T49" target="_self" title="BORSUK_4:th.49">Th49</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">E</font>
;<br/>

<b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">B</font></span>)</span><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green" title="E42">A3</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:47"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E7:47"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E8:47"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="topmetr.html#T31" title="TOPMETR:th.31">TOPMETR:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">e</font> =  <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">E</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e</font>;<br/>
<a NAME="E10:47"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="borsuk_4.html#E8:47"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T20" title="TREAL_1:th.20">TREAL_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">e</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">E</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">g</font>
;<br/>

<b>thus </b><a NAME="E12:47"/>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<b>thus </b><a NAME="E13:47"/>
<font color="Maroon" title="c12">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:47"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E10:47"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="tops_2.html#T71" title="TOPS_2:th.71">TOPS_2:71</a></i>;<br/>

<a NAME="E14:47"/><i><font color="Green" title="E45">A6</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="treal_1.html#D1" title="TREAL_1:def.1">TREAL_1:def 1</a></i>;<br/>
<a NAME="E15:47"/><i><font color="Green" title="E46">A7</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="treal_1.html#D2" title="TREAL_1:def.2">TREAL_1:def 2</a></i>;<br/>
<a NAME="E16:47"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E17:47"/><b>then </b>
<font color="Maroon" title="c5">a</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="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="borsuk_4.html#E6:47"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E14:47"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="treal_1.html#T16" title="TREAL_1:th.16">TREAL_1:16</a></i>
;<br/><br/>

<a NAME="E19:47"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E20:47"/><b>then </b>
<font color="Maroon" title="c6">b</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="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:47"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="borsuk_4.html#E6:47"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="borsuk_4.html#E15:47"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="treal_1.html#T16" title="TREAL_1:th.16">TREAL_1:16</a></i>
;<br/><br/>


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