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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">T0</font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">T0</font>;<br/>





<b>set </b><font color="Maroon">F</font> =  <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">TR</font> =  <a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font>;<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> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:25"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">C</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">C</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="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">C</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="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="t_0topsp.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:25_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/>
 ex <font color="Olive">h</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font> st <br/>for <font color="Olive">C</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">C</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="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>,<font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T0</font><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">C</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">C</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="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">C</font>
 ;<br/>
<a NAME="E5:25"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Olive">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E1:25_2"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:25_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:25_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">T0</font> ;<br/>
<a NAME="E7:25"/>
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T0</font>  st <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T0</font>;<br/>

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>set </b><font color="Maroon">V</font> = <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>;<br/>
<a NAME="E2:25_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_3_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:25_3_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:25_3_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, <a class="ref" href="t_0topsp.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:25_3_1_1"/><i><font color="Green">E68</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, , <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E8:25_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/><a NAME="E9:25_3_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E10:25_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:25_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:25_3_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:25_3_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="t_0topsp.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:25_3_1"/><b>then </b>
(  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E6:25_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25_3"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E4:25_3"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E5:25_3"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:25_3"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">T0</font> <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>set </b><font color="Maroon">H</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span>;<br/>
<a NAME="E9:25"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:25_4"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_4"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E4:25_4"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="t_0topsp.html#K2" target="_self">Indiscernible</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E5:25_4"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</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="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:25_4"/>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:25_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E8:25_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K1" target="_self">Indiscernibility</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:25_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:25_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:25"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E11:25"/>
 ex <font color="Olive">h</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">T0</font> st <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Olive">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>


</div>
