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

<b>assume </b><b>that </b><br/><a NAME="E1:60_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 <b>and </b><br/><a NAME="E2:60_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">A</font> being <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><br/> for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_5.html#D4" target="_self">TOPGEN_5:def 4</a><br/>

<b>thus </b><a NAME="E3:60_1_1"/>
<font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:60_1_1"><i><font color="Green">E43</font></i></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>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D5">COMPTS_1:def 5</a><br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E4:60_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E6:60_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = 0, <font color="Maroon">j</font> = 1, <font color="Maroon">half</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">half</font></span><a href="rcomp_2.html#K1">.[</a></span>, <font color="Maroon">B</font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">half</font>,<font color="Maroon">j</font></span><a href="rcomp_2.html#K2">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topreala.html#T11">TOPREALA:11</a>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topalg_1.html#T4">TOPALG_1:4</a>, <a class="ref" href="topalg_1.html#T5">TOPALG_1:5</a></i>;<br/>
<b>take </b><font color="Maroon">W</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>;<br/>

<b>take </b><font color="Maroon">V</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font>;<br/>

<b>thus </b><a NAME="E10:60_1_1"/>
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</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/>

<a NAME="E11:60_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<b>hence </b><a NAME="E12:60_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>

<a NAME="E13:60_1_1"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E14:60_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:60_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:60_1_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T163">FUNCT_1:163</a></i>;<br/>

<b>assume </b><a NAME="E17:60_1_1"/>
<font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:60_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E20:60_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="topgen_5.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/>
<a NAME="E22:60_1_1"/>
( <font color="Maroon">fx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">half</font> &amp; <font color="Maroon">fx</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">half</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E23:60_1_1"/>
contradiction
 ;<br/>


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