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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR4" target="_self">is_T4</a>  )
 ;<br/>

<b>hence </b><a NAME="E2:60_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_5.html#D4" target="_self">TOPGEN_5:def 4</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E3:60_2_1"/>
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> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 1, <font color="Maroon">c<sub>5</sub></font> =  <a href="real_1.html#K1">-</a> 1 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> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0, <font color="Maroon">c<sub>7</sub></font> = 1 as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E2:60_2_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:60_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:60_2_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="urysohn1.html#T27">URYSOHN1:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><a NAME="E4:60_2_1_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:60_2_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 1</span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T6">BORSUK_1:6</a></i>;<br/><b>assume </b><a NAME="E6:60_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E7:60_2_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E8:60_2_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topgen_5.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E10:60_2_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:60_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topgen_5.html#E8:60_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="boolmark.html#T3">BOOLMARK:3</a></i>;<br/><a NAME="E11:60_2_1_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</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="topgen_5.html#E5:60_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="txt" href="topgen_5.html#E11:60_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E13:60_2_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E11:60_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E15:60_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:60_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tietze.html#T25">TIETZE:25</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E17:60_2_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <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> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 ) &amp; <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T42">JGRAPH_5:42</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="tops_2.html#K2">/"</a>  as  <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> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><a NAME="E20:60_2_1_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E21:60_2_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E22:60_2_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topgen_5.html#E20:60_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="topgen_5.html#E15:60_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topgen_5.html#E12:60_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topgen_5.html#E13:60_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_5.html#E22:60_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><font color="Maroon">c<sub>16</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_5.html#E15:60_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topgen_5.html#E13:60_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/><a NAME="E25:60_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E10:60_2_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a>, <a class="txt" href="topgen_5.html#E21:60_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topgen_5.html#E17:60_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topgen_5.html#E20:60_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>;<br/><b>hence </b><a NAME="E26:60_2_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T13">SETWISEO:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
