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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="borsuk_4.html#E1:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="borsuk_4.html#T49" target="_self">Th49</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:71"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E7:71"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E8:71"/><b>then </b><i><font color="Green">E64</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="treal_1.html#K4">P[01]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E10:71"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="borsuk_4.html#E8:71"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T20">TREAL_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E12:71"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>thus </b><a NAME="E13:71"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="borsuk_4.html#E10:71"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>

<a NAME="E14:71"/><i><font color="Green">E66</font></i>: 
( 0 <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1">(#)</a> 0,1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1">(#)</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>;<br/>
<a NAME="E15:71"/><i><font color="Green">E67</font></i>: 
( 1 <a href="hidden.html#R1">=</a> 0,1 <a href="treal_1.html#K2">(#)</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="treal_1.html#K2">(#)</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<a NAME="E16:71"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E17:71"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="borsuk_4.html#E6:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="borsuk_4.html#E14:71"><i><font color="Green">E66</font></i></a>, <a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
;<br/><br/>

<a NAME="E19:71"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E20:71"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="borsuk_4.html#E6:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="borsuk_4.html#E15:71"><i><font color="Green">E67</font></i></a>, <a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
;<br/><br/>


</div>
