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

<b>let </b><font color="Maroon">FMT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> ;<br/>

<a NAME="E1:40"/><i><font color="Green">E19</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><br/> for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> ) ) implies for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><br/> for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> )
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<a NAME="E2:40_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font>;<br/>

<a NAME="E1:40_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<a NAME="E2:40_1_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">W1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  holds <br/>(  not <font color="Olive">W1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> or <font color="Olive">W1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> or <font color="Olive">W1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<b>assume </b><a NAME="E1:40_1_1_1_1"/>
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:40_1_1_1_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:40_1_1_1_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:40_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E5:40_1_1_1_1"/><b>then </b>
( <font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>hence </b><a NAME="E6:40_1_1_1_1"/>
( <font color="Maroon">W1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> or <font color="Maroon">W1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:40_1_1_1"/>
for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> &amp; not <font color="Olive">V1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> holds <br/><font color="Olive">V2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>, <font color="Maroon">V2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<b>assume </b><a NAME="E1:40_1_1_1_2"/>
( <font color="Maroon">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> &amp; <font color="Maroon">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font><b> such that </b><br/><a NAME="E3:40_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:40_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:40_1_1_1_2"/>
( <font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> or <font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:40_1"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E6:40_1_1_1_2"/><b>then </b>
( <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font>, <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><b> such that </b><br/><a NAME="E8:40_1_1_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> or <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E9:40_1_1_1_2"/>
( <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:40_1_1_1_2"/><b>then </b>
( <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:40_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:40_1_1_1_2"/><b>then </b>
( <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E12:40_1_1_1_2"/>
( <font color="Maroon">V1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> or <font color="Maroon">V2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:40_1_1_1"/><b>then </b>
( for <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> holds <br/><font color="Olive">V1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> or for <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font> holds <br/><font color="Olive">V2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E5:40_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 
;<br/>
<b>hence </b><a NAME="E6:40_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 ;<br/>

<a NAME="E2:40_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:40_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="fintopo2.html#E1:40"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/>
( ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> ) implies for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><br/> for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_2"/>
(  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> ex <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; ( for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> not <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> ) ) implies  ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font>, <font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>, <font color="Maroon">V2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font><b> such that </b><a NAME="E1:40_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x0</font> )
 <b>and </b><br/><a NAME="E2:40_2_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x0</font> holds <br/> not <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
 ;<br/>

<a NAME="E3:40_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_2_1_1"/>
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x0</font> holds <br/><font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>;<br/>

<b>assume </b><a NAME="E1:40_2_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E2:40_2_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
not <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
 
<b>by </b><i><a class="txt" href="fintopo2.html#E1:40"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E3:40_2_1_1_1"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_1_1_1"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:40_2_1_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2_1_1_1"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_2_1_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:40_2_1"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<a NAME="E2:40_2_1_2"/>
<font color="Maroon">V1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_2_1"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> 
 ;<br/>

<a NAME="E2:40_2_1_3"/>
<font color="Maroon">V2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> 
;<br/>

<b>take </b>
<font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> 
;<br/>

<b>thus </b><a NAME="E6:40_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_2"/>
( ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> ) implies for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><br/> for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font><br/> for <font color="Olive">V1</font>, <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font>  st <font color="Olive">V1</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">V2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">V2</font> ) ) iff for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> holds  <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font></span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 <b>by </b><i/>;<br/>


</div>
