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<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T2</font>;<br/><b>let </b><font color="Maroon">P1</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">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>, <font color="Maroon">P2</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">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>







<b>assume </b><a NAME="E1:33"/>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">P1</font>
 ;<br/>

<a NAME="E2:33"/><b>then </b>
<font color="Maroon">P2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>, <font color="Olive">Y1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> : ( <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D6">BORSUK_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">X11</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>, <font color="Maroon">Y11</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X11</font>,<font color="Maroon">Y11</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X11</font>,<font color="Maroon">Y11</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">X11</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y11</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>
<a NAME="E5:33"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P2</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"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X11</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y11</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_3.html#T94">FUNCT_3:94</a></i>;<br/>
<a NAME="E6:33"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X11</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y11</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/>
<b>hence </b><a NAME="E7:33"/>
<span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>


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