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<b>let </b><font color="Maroon">T</font> be   <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="E1:44"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">T</font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1"/>
(  not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>suppose </b></a><a NAME="E1:44_1_1"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:44_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="pre_topc.html#K4">Cl</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="hidden.html#R1">=</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#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <span class="p2">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:44_1_1"/>
 <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> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a>, <a class="ref" href="topgen_1.html#T1" target="_self">Th1</a>, <a class="ref" href="topgen_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:44_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="pre_topc.html#K4">Cl</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="tarski.html#R1">c=</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#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="topgen_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/><a NAME="E5:44_1_1"/>
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E6:44_1_1"/>
 <a href="pre_topc.html#K4">Cl</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="hidden.html#R1">=</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#D10">XBOOLE_0:def 10</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2"><b>suppose </b></a><a NAME="E1:44_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:44_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</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="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:44_1_2"/>
 <a href="pre_topc.html#K4">Cl</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="hidden.html#R1">=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="topgen_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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