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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<a NAME="E1:56"/><i><font color="Green">E34</font></i>: 
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">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">g</font> =  <a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> as  <a href="pre_topc.html#V5">continuous</a> <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">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V3">Hausdorff</a>
 ;<br/><b>let </b><font color="Maroon">A</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">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>assume </b><a NAME="E2:56_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E3:56_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:56_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:56"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">A</font> being  <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">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E5:56"/><i><font color="Green">E46</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <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">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> as   <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">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a><br/>

<b>assume </b><a NAME="E7:56"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E8:56"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:56"/><b>then </b>
<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">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">SF</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E11:56"/><i><font color="Green">E75</font></i>: 
<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">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">SF</font>
 <b>and </b><br/><a NAME="E12:56"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> holds <br/> ex <font color="Olive">X1</font>, <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <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> &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> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<a NAME="E13:56"/>
not <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E14:56"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</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">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">XY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:56"/><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">XY</font> &amp; <font color="Maroon">XY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E18:56"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">XY</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="yellow12.html#E1"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">X1</font>
;<br/>

<b>take </b>
<font color="Maroon">Y1</font>
;<br/>

<b>thus </b><a NAME="E19:56"/>
( <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E20:56"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<b>assume </b><a NAME="E21:56"/>
<font color="Maroon">X1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:56"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E24:56"/>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:56"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">XY</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E26:56"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">SF</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E27:56"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E28:56"/>
contradiction
 <b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div>
