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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="E1:52"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  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#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:52_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#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_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="E2:52_1_1"/><b>then </b>
<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"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:52_1_1"/><b>then </b>
<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="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:52_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#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>


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

<b>assume </b><a NAME="E2:52_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#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span>
 ;<br/>

<a NAME="E3:52_1"/><b>then </b>
<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="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b>
<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"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:52_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> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:52"/>
<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#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fintopo2.html#K10" target="_self">^Foi</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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