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<b>let </b><font color="Maroon">T</font> be  <a href="t_0topsp.html#V2">T_0</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">T</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E2:40"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:40"/><i><font color="Green">E32</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E5:40"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">B1</font></span>)</span> <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/>;<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="E7:40"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow15.html#K2" target="_self">Components</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E8:40"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">infinite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:40"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:40"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E14:40"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E16:40"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E17:40"/><b>then </b>
not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E19:40"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E20:40"/><i><font color="Green">E53</font></i>: 
( ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) or ( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i>, <a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:40_1_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E4:40_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:40_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:40_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><a NAME="E4:40_1_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:40_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E22:40"/>
contradiction
 ;<br/>


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