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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="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  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="E1:27"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <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>
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font> <a href="borsuk_3.html#R2" target="_self">are_homeomorphic</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:27"/><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">Z</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_3.html#R2" target="_self">are_homeomorphic</a> 
 
<b>by </b><i><a class="ref" href="yellow12.html#T44">YELLOW12:44</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="borsuk_3.html#T1" target="_self">Th1</a>, <a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</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">Z</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Z</font></span>)</span>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E7:27"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">gf</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</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">Z</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font> ;<br/>
<a NAME="E9:27"/>
<font color="Maroon">gf</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E2:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>hence </b><a NAME="E10:27"/>
<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">Z</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">Y</font> <a href="borsuk_3.html#R2" target="_self">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>


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