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

<b>let </b><font color="Maroon">g1</font> be  <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>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">S</font></span>)</span>, <font color="Maroon">g2</font> be  <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>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">S</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g1</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>and </b><br/><a NAME="E2:32_1_2"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="t_1topsp.html#K5" target="_self">T_1-reflect</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g2</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>
 ;<br/>

<a NAME="E3:32_1_2"/><i><font color="Green">E22</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</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;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>now </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="E1:32_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 ;<br/><a NAME="E2:32_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<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="E3:32_1_2_1"/><i><font color="Green">E24</font></i>: 
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> <a href="hidden.html#R1">=</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="E5:32_1_2_1"/><i><font color="Green">E25</font></i>: 
<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><a class="txt" href="t_1topsp.html#E5:32_1_2_1"><i><font color="Green">E25</font></i></a>, </i>;<br/><a NAME="E6:32_1_2_1"/><i><font color="Green">E26</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>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/><a NAME="E8:32_1_2_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="E9:32_1_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">y</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">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="t_1topsp.html#E9:32_1_2_1"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/><b>set </b><font color="Maroon">ty</font> = <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">y</font>;<br/><a NAME="E10:32_1_2_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">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="txt" href="t_1topsp.html#E6:32_1_2_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E11:32_1_2_1"/><b>then </b><i><font color="Green">E57</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_2.html#K3">.</a> <font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">x</font> or <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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="t_1topsp.html#E5:32_1_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="t_1topsp.html#E6:32_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>hence </b><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</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="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="t_1topsp.html#E3:32_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="t_1topsp.html#E2:32_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a>, , <a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_1_2"/>
<font color="Maroon">g1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
