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



<a NAME="E1:49"/>
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> <font color="Maroon">A</font> <a href="fintopo2.html#K8" target="_self">^Fodelta</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#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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/>

<b>thus </b><a NAME="E1:49_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K8" target="_self">^Fodelta</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#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K8" target="_self">^Fodelta</a> 
 ;<br/>

<a NAME="E2:49_1_1"/><b>then </b>
( ( 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">x</font> holds <br/><font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</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="Maroon">x</font> holds <br/><font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:49_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>  &amp; <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#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 
;<br/>
<b>hence </b><a NAME="E4:49_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#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</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#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:49_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#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a> </span>)</span>
 ;<br/>

<a NAME="E3:49_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>  &amp; <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#K3">`</a> </span>)</span> <a href="fintopo2.html#K9" target="_self">^Fob</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:49_1"/><b>then </b>
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">x</font> holds <br/>( <font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:49_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K8" target="_self">^Fodelta</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:49"/>
<font color="Maroon">A</font> <a href="fintopo2.html#K8" target="_self">^Fodelta</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#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</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="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


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