<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">FMT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="fintopo2.html#L1" title="FINTOPO2:struct.1">FMT_Space_Str</a> ;<br/>

<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">FMT</font> is <a href="fintopo2.html#V2" title="FINTOPO2:attr.2">Fo_filled</a>
 ;<br/>

<a NAME="E2:43"/>
( ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> ) implies for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1"/>
(  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font> ex <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b2">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; ( for <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) ) implies  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">x0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font>, <font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font><b> such that </b><a NAME="E2:43_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c4">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">x0</font> )
 <b>and </b><br/><a NAME="E3:43_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">x0</font> holds <br/> not <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font>
 ;<br/>

<a NAME="E4:43_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_1_1"/>
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">x0</font> holds <br/>( <font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>;<br/>

<b>assume </b><a NAME="E1:43_1_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c2">x0</font>
 ;<br/>

<a NAME="E2:43_1_1_1_1"/><b>then </b><i><font color="Green" title="E22">A6</font></i>: 
not <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E3:43_1_1"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<a NAME="E3:43_1_1_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_1_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<a NAME="E2:43_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo2.html#E2:43_1_1_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:43_1_1_1_1"/>
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V2</font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:43"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fintopo2.html#E2:43_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="fintopo2.html#D8" target="_self" title="FINTOPO2:def.8">Def8</a></i>;<br/>
<a NAME="E5:43_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:43_1_1_1_1"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:43"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fintopo2.html#E1:43_1_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="fintopo2.html#D8" target="_self" title="FINTOPO2:def.8">Def8</a></i>;<br/>
<a NAME="E7:43_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <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> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="fintopo2.html#E5:43_1_1_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:43_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V2</font></span>)</span> <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> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1_1_1"/>
( <font color="Maroon" title="c5">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c5">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="fintopo2.html#E3:43_1_1_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1_1_1"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:43_1_1"/><i><font color="Green" title="E21">A9</font></i>: 
not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 ;<br/>

<a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">V1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="fintopo2.html#E2:43_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="fintopo2.html#T20" target="_self" title="FINTOPO2:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:43_1_1"/><i><font color="Green" title="E22">A10</font></i>: 
not <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_3"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> 
 ;<br/>

<a NAME="E2:43_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">V2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="fintopo2.html#E2:43_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="fintopo2.html#T20" target="_self" title="FINTOPO2:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
;<br/>

<b>take </b>
<font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
;<br/>

<b>thus </b><a NAME="E7:43_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">V2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fintopo2.html#E4:43_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="fintopo2.html#E5:43_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="fintopo2.html#E6:43_1_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1"/>
( ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> ) implies for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
(  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st not <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="fintopo2.html#K8" title="FINTOPO2:func.8">^Fodelta</a> </span>)</span> or for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FMT</font><br/> for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font>  st <font color="Olive" title="b2">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FMT</font> st <br/>( <font color="Olive" title="b4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">V2</font> ) )
 ;<br/>


</div>
