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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</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">w1</font>, <font color="Maroon" title="c4">w2</font>, <font color="Maroon" title="c5">w3</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>assume </b><a NAME="E1:48"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c2">P</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> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1"/>
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</font> or <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>suppose </b></a><a NAME="E1:48_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:48_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> &amp; <font color="Maroon" title="c7">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">w2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:48_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:48_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:48_1_1"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:48_1_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:48_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:48_1_1"/><b>then </b><i><font color="Green" title="E46">A7</font></i>: 
<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/><a NAME="E10:48_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</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">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<font color="Maroon" title="c2">P</font> <b>by </b><i><a class="txt" href="jordan2c.html#E7:48_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:48_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">gt</font> = <font color="Maroon" title="c8">g</font> as   <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="c6">Y</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_1_1"/>
( <font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w3</font> or <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1"/>
<font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w3</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">f2</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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:48_1_1_1_1_1"/><i><font color="Green" title="E47">A8</font></i>: 
( <font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> &amp; <font color="Maroon" title="c10">f2</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="c5">w3</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:48_1_1_1_1_1"/><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c10">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:48_1_1_1_1_1"/><i><font color="Green" title="E49">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:48_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E50">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:48_1_1_1_1_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:48_1_1_1_1_1"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:48_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A12</font></i>: 
<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/><a NAME="E10:48_1_1_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f2</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">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">g2</font> = <font color="Maroon" title="c10">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<font color="Maroon" title="c2">P</font> <b>by </b><i><a class="txt" href="jordan2c.html#E7:48_1_1_1_1_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:48_1_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">gt2</font> = <font color="Maroon" title="c11">g2</font> as   <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="c6">Y</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E14:48_1_1_1_1_1"/><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c12">gt2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:48_1_1_1_1_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="jordan2c.html#E9:48_1_1_1_1_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><a NAME="E15:48_1_1_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">w1'</font> = <font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c14">w2'</font> = <font color="Maroon" title="c4">w2</font>, <font color="Maroon" title="c15">w3'</font> = <font color="Maroon" title="c5">w3</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/><a NAME="E17:48_1_1_1_1_1"/>
( <font color="Maroon" title="c9">gt</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c13">w1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">gt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c14">w2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">gt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="jordan2c.html#E5:48_1_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="jordan2c.html#E9:48_1_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><a NAME="E18:48_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">h</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="c6">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c13">w1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c15">w3'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">gt</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">gt2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="jordan2c.html#E14:48_1_1_1_1_1"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="borsuk_2.html#T16" title="BORSUK_2:th.16">BORSUK_2:16</a></i>;<br/><b>hence </b><a NAME="E19:48_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w3</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:48_1_1_1_1_2"/><i><font color="Green" title="E48">A15</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> &amp; <font color="Maroon" title="c10">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="c5">w3</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:48_1_1_1_1_2"/><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:48_1_1_1_1_2"/><i><font color="Green" title="E50">A17</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>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:48_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A18</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan2c.html#E1:48_1_1_1_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:48_1_1_1_1_2"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:48_1_1_1_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:48_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A19</font></i>: 
<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/><a NAME="E10:48_1_1_1_1_2"/>
 <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> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<font color="Maroon" title="c2">P</font> <b>by </b><i><a class="txt" href="jordan2c.html#E7:48_1_1_1_1_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:48_1_1_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">gt</font> = <font color="Maroon" title="c11">g</font> as   <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="c6">Y</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E14:48_1_1_1_1_2"/>
<font color="Maroon" title="c12">gt</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:48_1_1_1_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="jordan2c.html#E9:48_1_1_1_1_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><b>hence </b><a NAME="E15:48_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_1_1_1_2"><i><font color="Green" title="E48">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:48_1_1"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><b>suppose </b></a><a NAME="E1:48_1_2"/><i><font color="Green" title="E41">A20</font></i>: 
<font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_1_1"/>
( <font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w3</font> or <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_1"><b>case </b></a><a NAME="E1:48_1_2_1_1_1"/>
<font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w3</font>
 ;<br/><div class="add"><a NAME="E2:48_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48_1_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:48_1_2_1_1_1"/><i><font color="Green" title="E42">A21</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> &amp; <font color="Maroon" title="c7">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="c5">w3</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:48_1_2_1_1_1"/><i><font color="Green" title="E43">A22</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_2_1_1_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:48_1_2_1_1_1"/><i><font color="Green" title="E44">A23</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_2_1_1_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:48_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="jordan2c.html#E1:48_1_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E8:48_1_2_1_1_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:48_1_2_1_1_1"><i><font color="Green" title="E44">A23</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E9:48_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E46">A25</font></i>: 
<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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">w3</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/><a NAME="E10:48_1_2_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</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">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>,<font color="Maroon" title="c2">P</font> <b>by </b><i><a class="txt" href="jordan2c.html#E7:48_1_2_1_1_1"><i><font color="Green" title="E45">A24</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:48_1_2_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">gt</font> = <font color="Maroon" title="c8">g</font> as   <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="c6">Y</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><a NAME="E14:48_1_2_1_1_1"/>
<font color="Maroon" title="c9">gt</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:48_1_2_1_1_1"><i><font color="Green" title="E43">A22</font></i></a>, <a class="txt" href="jordan2c.html#E9:48_1_2_1_1_1"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/><b>hence </b><a NAME="E15:48_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:48_1_2_1_1_1"><i><font color="Green" title="E42">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_2"><b>case </b></a><a NAME="E1:48_1_2_1_1_2"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w3</font>
 ;<br/><div class="add"><a NAME="E2:48_1_2_1_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">w1'</font> = <font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c8">w3'</font> = <font color="Maroon" title="c5">w3</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <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">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/><a NAME="E4:48_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">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="c6">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1'</font> &amp; <font color="Olive" title="b1">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="c8">w3'</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:48_1_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="jordan2c.html#E1:48_1_2_1_1_2"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="borsuk_2.html#T4" title="BORSUK_2:th.4">BORSUK_2:4</a></i>;<br/><b>hence </b><a NAME="E5:48_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_2"/>
 ex <font color="Olive" title="b1">h</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="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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