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<b>let </b><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c5">Z</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">T</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>;<br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c5">Z</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 ;<br/>

<a NAME="E3:24"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b2">P</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">p</font>  ex <font color="Olive" title="b3">R</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> st <font color="Olive" title="b3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">S</font>;<br/><b>let </b><font color="Maroon" title="c9">P</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">p</font>;<br/>



<b>consider </b><font color="Maroon" title="c10">R</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font><b> such that </b><br/><a NAME="E2:24_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c10">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="topreala.html#E2:24"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="topgrp_1.html#T22" title="TOPGRP_1:th.22">TOPGRP_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">R2</font> = <font color="Maroon" title="c10">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">Z</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">Z</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">R2</font> = <font color="Maroon" title="c11">R2</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <b>by </b><i><a class="txt" href="topreala.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topreala.html#T26" target="_self" title="TOPREALA:th.26">Th26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">R2</font>
;<br/>

<a NAME="E5:24_1"/>
<font color="Maroon" title="c12">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1"/>
<font color="Maroon" title="c12">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">P</font>
 <b>by </b><i><a class="txt" href="topreala.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topreala.html#E2:24_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon" title="c7">g</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T23" title="TOPGRP_1:th.23">TOPGRP_1:23</a></i>;<br/>


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