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<b>let </b><font color="Maroon" title="c2">X</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U</font>, <font color="Olive" title="b2">V</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" title="TOPGEN_3:func.5">ClFinTop</a> <font color="Maroon" title="c2">X</font></span>)</span>  holds <font color="Olive" title="b1">U</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Olive" title="b2">V</font></span><br/><b>let </b><font color="Maroon" title="c3">U</font>, <font color="Maroon" title="c4">V</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" title="TOPGEN_3:func.5">ClFinTop</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">U</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c4">V</font></span><br/>



<b>assume </b><a NAME="E1:50"/>
<font color="Maroon" title="c3">U</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c4">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:50"/><b>then </b>
( <font color="Maroon" title="c3">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c4">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#T34" target="_self" title="TOPGEN_3:th.34">Th34</a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E3:50"/><b>then </b><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c3">U</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" title="TOPGEN_3:func.5">ClFinTop</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#D6" target="_self" title="TOPGEN_3:def.6">Def6</a></i>;<br/>
<a NAME="E4:50"/><b>then </b>
not <font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T35" target="_self" title="TOPGEN_3:th.35">Th35</a></i>;<br/>
<a NAME="E5:50"/><b>then </b>
<font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" title="TOPGEN_3:func.5">ClFinTop</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="topgen_3.html#D6" target="_self" title="TOPGEN_3:def.6">Def6</a></i>;<br/>
<a NAME="E6:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K5" title="TOPGEN_3:func.5">ClFinTop</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E7:50"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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