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<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/>



<a NAME="E1:29"/><i><font color="Green">E19</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/>
<b>assume </b><a NAME="E2:29"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">T1</font> is <a href="urysohn1.html#V1">being_T1</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E3:29"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">w</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>
 ;<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="E4:29"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">w</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">fz</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">z</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T1</font> : <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> </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><a class="txt" href="t_1topsp.html#E2:29"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E6:29"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fz</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E2:29"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E7:29"/><b>then </b>
<font color="Maroon">fz</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon">fz</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is    <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> : <font color="Olive">B</font> is <a href="tops_2.html#V2">closed</a> </span>}</span> 
 
;<br/>
<a NAME="E9:29"/><b>then </b>
<font color="Maroon">fz</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">fz</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <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> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span> 
 
;<br/>
<a NAME="E11:29"/><i><font color="Green">E25</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="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">fz</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</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"><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>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:29_1"/><i><font color="Green">E27</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fx</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="E5:29_1"/>
<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">fx</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fz</font>
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E3:29_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:29_1"/>
<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>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">fz</font>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E2:29_1"><i><font color="Green">E26</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:29"/>
 <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="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <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> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <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="t_1topsp.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E13:29"/>
<font color="Maroon">y</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"><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>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E3:29"><i><font color="Green">E22</font></i></a>, <a class="txt" href="t_1topsp.html#E4:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="t_1topsp.html#E10:29"><i><font color="Green">E24</font></i></a>, <a class="txt" href="t_1topsp.html#E11:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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