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<b>let </b><font color="Maroon">X</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">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>







<b>reconsider </b><font color="Maroon">V</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<b>assume </b><a NAME="E2:31"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<a NAME="E3:31"/>
<font color="Maroon">Y</font>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_3.html#R2" target="_self">are_homeomorphic</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:31"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span> #)
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">Z</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, , <a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>


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