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<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1" target="_self">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 

;<br/>
<i><font color="Green">E33</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:16"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:16"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">s3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:16"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font> &amp; <font color="Maroon">s3</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="jordan5c.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E12:16"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:16"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:16"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">s4</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:16"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">s4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s4</font> &amp; <font color="Maroon">s4</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma16</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E18:16"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s4</font>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#T1" target="_self">Th1</a>, , , , </i>;<br/>
<a NAME="E19:16"/>
<font color="Maroon">s4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s3</font>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#T1" target="_self">Th1</a>, , , , </i>;<br/>
<b>hence </b><a NAME="E20:16"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="jordan5c.html#T1" target="_self">Th1</a>, , , , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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