<?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="urysohn1.html#V1">being_T1</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</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>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </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> )
 ;<br/>

<b>thus </b><a NAME="E2:66"/>
<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>3</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>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E3:66"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E4:66"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:66"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:66"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:66"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 0, <font color="Maroon">c<sub>8</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="66_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>6</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="66_1_1"><b>suppose </b></a><a NAME="E1:66_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:66_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:66"><i><font color="Green">E50</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E3:66_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E6:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:66_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>7</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>7</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T8">TOPS_1:8</a></i>;<br/><a NAME="E5:66_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>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <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; <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>7</sub></font></span>)</span> <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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:66_1_1"/>
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>4</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>3</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="66_1_2"><b>suppose </b></a><a NAME="E1:66_1_2"/>
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_5.html#E8:66"><i><font color="Green">E50</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</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="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </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> );<br/><a NAME="E3:66_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:66_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E6:66"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_5.html#E8:66"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topgen_5.html#E1:66_1_2_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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> <b> such that </b><br/><a NAME="E5:66_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K3">`</a> </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> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:66"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/>

<b>thus </b><a NAME="E6:66_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font>]
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:66_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="topgen_5.html#E3:66_1_2"><i><font color="Green">E51</font></i></a>);<br/></i><a NAME="E6:66_1_2"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:66_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:66_1_2_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>]
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:66_1_2_3"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E3:66_1_2_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is  <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> 
 ;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>12</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> <b> such that </b><br/><a NAME="E10:66_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T55" target="_self">Th55</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="topmetr.html#K5">I[01]</a> ;<br/><a NAME="E13:66_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_1_2_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:66_1_2_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 
<b>by </b><i><a class="txt" href="topgen_5.html#E3:66_1_2_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:66_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>9</sub></font>, <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E15:66_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E16:66_1_2"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E19:66_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:66_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_2_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_2_5_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:66_1_2_5_1"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>] &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:66_1_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E15:66_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>
<b>hence </b><a NAME="E4:66_1_2_5_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_5.html#E19:66_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:66_1_2_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:66_1_2_5"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:66_1_2_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>]
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topgen_5.html#E4:66_1_2_5"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E6:66_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="topgen_5.html#E15:66_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topgen_5.html#E4:66_1_2_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>
<b>hence </b><a NAME="E7:66_1_2_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="topgen_5.html#E10:66_1_2"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>
;<br/><br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E22:66_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:66_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_5.html#E24:66_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E27:66_1_2"/><i><font color="Green">E57</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</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="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/><a NAME="E29:66_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#NK1" target="_self">max</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E10:66_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E30:66_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topgen_5.html#E27:66_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:66_1_2_7"/><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = 1 as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E3:66_1_2_7"/>
not <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E24:66_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_5.html#E6:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:66_1_2_7"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:66"><i><font color="Green">E50</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>28</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> <b> such that </b><br/><a NAME="E7:66_1_2_7"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>28</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="subset_1.html#K3">`</a> </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> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topgen_5.html#E5:66_1_2_7"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E8:66_1_2_7"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2_7"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topgen_5.html#E7:66_1_2_7"><i><font color="Green">E62</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E9:66_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E10:66_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topgen_5.html#E7:66_1_2_7"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:66_1_2_7"/><b>then </b>
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:66_1_2_7"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topgen_5.html#E7:66_1_2_7"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topgen_5.html#E15:66_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/><b>hence </b><a NAME="E12:66_1_2_7"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>thus </b><a NAME="E13:66_1_2_7"/>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:66_1_2_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/></div><b>end;</b></div><a NAME="E32:66_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="topgen_5.html#E24:66_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topgen_5.html#E29:66_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topgen_5.html#E30:66_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E33:66_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
