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<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>;<br/>





<b>assume </b><a NAME="E1:18"/>
<font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:18"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:18"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">t</font></span>)</span> where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">s</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b2">t</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:18"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="substlat.html#D3" title="SUBSTLAT:def.3">SUBSTLAT:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E6:18"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">t1</font> &amp; <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">s1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c6">t1</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">f1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E8:18"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font> &amp; ( for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> not <font color="Maroon" title="c7">f1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b1">g</font> ) )
 <b>by </b><i><a class="txt" href="heyting2.html#E6:18"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:18"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting2.html#E6:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="heyting2.html#E8:18"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>


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