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

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>;<br/>
<i><font color="Green">E13</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
0,1 <a href="treal_1.html#K2">(#)</a> 
<b>by </b><i><a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
;<br/>
<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
 <a href="treal_1.html#K1">(#)</a> 0,1
<b>by </b><i><a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E4:26_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E2:26_1_1_3"/><i><font color="Green">E16</font></i>: 
( 0,1 <a href="treal_1.html#K2">(#)</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="treal_1.html#K1">(#)</a> 0,1 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<a NAME="E3:26_1_1_3"/>
<font color="Maroon">c<sub>7</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#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E4:26_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:26_1_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="treal_1.html#T10">TREAL_1:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E6:26_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:26_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E6:26_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E7:26_1_1"/><b>then </b>
0 <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#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E8:26_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0 = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:26_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E10:26_1_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E11:26_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E12:26_1_1"/><b>then </b>
1 <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#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E13:26_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E17</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:26_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E15:26_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E16:26_1_1"/>
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T11">TREAL_1:11</a></i>;<br/>
<a NAME="E17:26_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E15:26_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E18:26_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="borsuk_2.html#R2" target="_self">are_connected</a> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E9:26_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="borsuk_2.html#E14:26_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="borsuk_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="borsuk_2.html#M1" target="_self">Path</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="borsuk_2.html#E9:26_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="borsuk_2.html#E14:26_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="borsuk_2.html#E17:26_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E20:26_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E4:26_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
