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





<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K11" target="_self">^Fos</a>  implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</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">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K11" target="_self">^Fos</a> 
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">y</font> &amp; <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) ) </span>}</span> 
 
;<br/>
<a NAME="E3:31_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">y</font> &amp; <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) )
 
;<br/>
<b>hence </b><a NAME="E4:31_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</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">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) )
 ;<br/>


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

<b>assume </b><a NAME="E2:31"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</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">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) )
 ;<br/>

<a NAME="E3:31"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FMT</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FMT</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">y</font> &amp; <font color="Olive">V</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> ) ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fintopo2.html#K11" target="_self">^Fos</a> 
 ;<br/>


</div>
