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<b>let </b><font color="Maroon" title="c1">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> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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> 2</span>)</span>;<br/>









<b>given </b><font color="Maroon" title="c6">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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">A</font></span>)</span><b> such that </b><a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c6">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="c2">p1</font>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c6">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="c3">p2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a><br/>

<b>assume </b><a NAME="E4:19"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/>

<a NAME="E5:19"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E4:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<i><font color="Green" title="E19">A7</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<b>by </b><i><a class="txt" href="jordan16.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<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></i>
;<br/>
<i><font color="Green" title="E20">A8</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan16.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</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 consider </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:19"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c7">u</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="c6">f</font>
 <b>and </b><br/><a NAME="E11:19"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E5:19"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s1</font> = <font color="Maroon" title="c7">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E10:19"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:19"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c9">u</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="c6">f</font>
 <b>and </b><br/><a NAME="E15:19"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E6:19"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan16.html#E8:19"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">s2</font> = <font color="Maroon" title="c9">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E14:19"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">s1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">s2</font>
;<br/>

<b>thus </b><a NAME="E17:19"/>
( <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c6">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="c2">p1</font> &amp; <font color="Maroon" title="c6">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="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan16.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan16.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/>

<b>thus </b><a NAME="E18:19"/>
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E11:19"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="jordan16.html#E15:19"><i><font color="Green" title="E24">A12</font></i></a></i>;<br/>

<a NAME="E19:19"/><i><font color="Green" title="E25">A13</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s1</font> &amp; <font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E10:19"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>thus </b><a NAME="E20:19"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E10:19"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>

<a NAME="E21:19"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font> &amp; <font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E14:19"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E22:19"/>
<font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan16.html#E2:19"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan16.html#E3:19"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan16.html#E4:19"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan16.html#E11:19"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="jordan16.html#E15:19"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="jordan16.html#E19:19"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>

<b>thus </b><a NAME="E23:19"/>
<font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan16.html#E7:19"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan16.html#E14:19"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


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