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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/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">T</font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<b>assume </b><a NAME="E4:43"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">G</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:43"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> or  not <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> ) ) )
 <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:43"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="topgen_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="topgen_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:43"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:43"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a>, , <a class="ref" href="topgen_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:43"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">G</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon">g</font> <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/>
<a NAME="E5:43_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:43_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E7:43_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:43"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="topgen_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E15:43"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E16:43"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</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/>
<a NAME="E17:43"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E18:43"/>
 <a href="pre_topc.html#K4">Cl</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> <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>
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#T35">YELLOW_8:35</a></i>;<br/>
<a NAME="E19:43"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</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> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="frechet.html#T4">FRECHET:4</a></i>;<br/>
<b>set </b><font color="Maroon">U</font> = <font color="Maroon">G</font> <a href="subset_1.html#K7">\</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/>
<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E21:43"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <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>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="txt" href="topgen_1.html#E1:43"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E22:43"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</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> )
 
<b>by </b><i><a class="txt" href="topgen_1.html#E2:43"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:43"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E2:43"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E25:43"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</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> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E26:43"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> )
 
<b>by </b><i><a class="txt" href="topgen_1.html#E4:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E27:43"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E4:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E28:43"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</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>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E29:43"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div>
