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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:76_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>

<a NAME="E2:76_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T70" target="_self" title="SETWISEO:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E3:76_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T68" target="_self" title="SETWISEO:th.68">Th68</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:76_1"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self" title="SETWISEO:th.14">Th14</a></i>;<br/><a NAME="E4:76_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x'</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T68" target="_self" title="SETWISEO:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E5:76_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:76_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="setwiseo.html#T70" target="_self" title="SETWISEO:th.70">Th70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:76"/>
 <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="setwiseo.html#K11" title="SETWISEO:func.11">singleton</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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