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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st ( for <font color="Olive" title="b3">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b2">p</font><br/> for <font color="Olive" title="b4">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b4">Q</font> ) holds <br/> ex <font color="Olive" title="b3">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b2">p</font> st <br/>for <font color="Olive" title="b4">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b4">Q</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <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">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st ( for <font color="Olive" title="b2">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b1">p</font><br/> for <font color="Olive" title="b3">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b3">Q</font> ) holds <br/> ex <font color="Olive" title="b2">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b1">p</font> st <br/>for <font color="Olive" title="b3">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b3">Q</font></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <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">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font><br/> for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Q</font> ) implies  ex <font color="Olive" title="b1">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font> st <br/>for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Q</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E13">A1</font></i>: 
for <font color="Olive" title="b1">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font><br/> for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Q</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font>  ex <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c4">K</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font>;<br/>
<a NAME="E4:31"/>
 ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Q</font> )
 
<b>by </b><i><a class="txt" href="yellow13.html#E2:31"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow13.html#E1:31"><i><font color="Green" title="E13">A1</font></i></a></i>; <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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