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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#L1" title="PRE_TOPC:struct.1">TopStruct</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="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>  &amp; <font color="Olive" title="b2">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="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>  &amp; <font color="Olive" title="b1">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be   <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>  &amp; <font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><br/>

<a NAME="E3:1"/>
<font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="jordan18.html#E1:1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/>
<a NAME="E4:1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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="c1">S</font> )
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E5:1"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan18.html#E2:1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="compts_1.html#T24" title="COMPTS_1:th.24">COMPTS_1:24</a></i>;<br/>
<a NAME="E6:1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</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> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="jordan18.html#E1:1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:1"/>
<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="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="jordan18.html#E5:1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="tops_2.html#T68" title="TOPS_2:th.68">TOPS_2:68</a></i>; <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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