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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">A</font> being <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/>

<a NAME="E2:37"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a></i>;<br/>
<b>thus </b> <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:37"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="tops_2.html#T73" title="TOPS_2:th.73">TOPS_2:73</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:37"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="tops_2.html#T52" title="TOPS_2:th.52">TOPS_2:52</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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