<?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>, <font color="Maroon">T1</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">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">T1</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">T1</font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">fx</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font>;<br/>
<a NAME="E3:31"/><i><font color="Green">E21</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green">E22</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</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> ] <b>means </b>for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E5:31"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">w</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">w</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">v</font>] holds <br/><font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">y</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> &amp; ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> ) )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">y</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="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:31_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">fix</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E8:31_1"/>
( <font color="Maroon">fix</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E4:31"><i><font color="Green">E22</font></i></a>, <a class="txt" href="t_1topsp.html#E5:31"><i><font color="Green">E23</font></i></a>, , <a class="txt" href="t_1topsp.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:31_1"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E6:31"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> holds <br/> ex <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">y</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<a NAME="E2:31_2"/><b>then </b>
<font color="Maroon">y</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="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:31_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:31_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>set </b><font color="Maroon">w</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><a NAME="E7:31_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E8:31_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font>
 
;<br/>
<a NAME="E9:31_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E5:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fix</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E11:31_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E12:31_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E13:31_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E4:31"><i><font color="Green">E22</font></i></a>, <a class="txt" href="t_1topsp.html#E5:31_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a>, <a class="txt" href="t_1topsp.html#E1:31"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E14:31_2"/>
not <font color="Maroon">y</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E5:31_2"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:31_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E17:31_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</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">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:31"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="txt" href="t_1topsp.html#E6:31"><i><font color="Green">E26</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>;<br/>
<a NAME="E9:31"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">w</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">w</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">v</font>] holds <br/><font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:31_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) &amp; ( for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">z</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:31_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/>
<a NAME="E4:31_3"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> )
 
<b>by </b><i>, <a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:31_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:31"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> holds <br/> ex <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:31_4"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><a NAME="E4:31_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_4_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 ;<br/><b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_4_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:31_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></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/><a NAME="E5:31_4_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E6:31_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:31_4_1"/>
contradiction
 <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><a NAME="E6:31_4"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:31"/><i><font color="Green">E74</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a>, );<br/></i>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font>;<br/>
<a NAME="E13:31"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_5"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">z</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="E2:31_5"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">z</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E4:31_5"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="t_1topsp.html#E4:31"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E5:31_5"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:31_5"/><b>then </b>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E7:31_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:31"/><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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>thus </b><a NAME="E1:31_6"/>
(  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:31"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:31_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_7"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E5:31_7"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E6:31_7"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, , </i>;<br/>
<b>hence </b><a NAME="E7:31_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:31"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:31"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_8"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">y</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="E2:31_8"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</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="E3:31_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:31_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="t_1topsp.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:31"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_9"><b>proof </b></a><div class="add">

<a NAME="E1:31_9"/>
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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</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="31_9_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/>

<b>assume </b><a NAME="E1:31_9_1"/><i><font color="Green">E85</font></i>: 
<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>
 ;<br/>

<a NAME="E2:31_9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:31_9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 
;<br/>
<a NAME="E4:31_9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <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="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:31_9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:31_9_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</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="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E9:31_9_1"/>
 <a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font> <a href="funct_2.html#R1">=</a>  <a href="borsuk_1.html#K2">proj</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D11">BORSUK_1:def 11</a></i>;<br/>
<a NAME="E10:31_9_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/>
<a NAME="E11:31_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:31_9_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fix</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font> ;<br/>
<a NAME="E14:31_9_1"/>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T6" target="_self">Th6</a>, , </i>;<br/>
<a NAME="E15:31_9_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">fix</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E4:31"><i><font color="Green">E22</font></i></a>, , <a class="txt" href="t_1topsp.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:31_9_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font>
 
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">fix</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="t_1topsp.html#T7" target="_self">Th7</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, <a class="ref" href="t_1topsp.html#T7" target="_self">Th7</a>, <a class="ref" href="t_1topsp.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E18:31_9_1"/>
<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"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_9"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E5:31"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h1</font> as   <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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">T1</font> ;<br/>
<a NAME="E21:31"/>
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E1:31_10"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">y</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">hy</font> = <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E3:31_10"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</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="E1:31_10_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">hy</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_10_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">hy</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:31_10_1"/>
<font color="Maroon">z2</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="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:31_10_1"/><b>then </b>
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:31_10_1"/>
<font color="Maroon">z1</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_1topsp.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">uhy</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">hy</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E5:31_10"/>
<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_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1" target="_self">Def1</a>, , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E6:31_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T181">RELAT_1:181</a></i>;<br/>
<a NAME="E7:31_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15">Proj</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">hy</font> is <a href="pre_topc.html#V4">closed</a>
 
;<br/>
<a NAME="E8:31_10"/><b>then </b>
<font color="Maroon">uhy</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:31_10"/>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font> is <a href="pre_topc.html#V4">closed</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_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>,<font color="Maroon">T1</font> ;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E23:31"/>
<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_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="t_1topsp.html#D1" target="_self">Def1</a></i>;<br/>


</div>
