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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E3:34_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>thus </b><a NAME="E4:34_1"/><i><font color="Green">E49</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">P</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>
;<br/><b>hence </b><a NAME="E3:34_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</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/></div>
<b>end;</b></div><b>assume </b><a NAME="E6:34_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E7:34_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> 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="E8:34_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/><a NAME="E9:34_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><b>hence </b><a NAME="E10:34_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="topgrp_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:34"/><i><font color="Green">E54</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> iff <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:34_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/><b>hence </b><a NAME="E3:34_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font>
<b>by </b><i>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>
;<br/><b>hence </b><a NAME="E3:34_3"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:34"/><b>then </b>
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E9:34"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>


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