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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">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">L</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>let </b><font color="Maroon" title="c6">a</font> be    <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>;<br/>











<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c5">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E2:42"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:42"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c6">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="heyting2.html#T2" target="_self" title="HEYTING2:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K8" title="HEYTING2:func.8">Atom</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E3:42"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="heyting2.html#T26" target="_self" title="HEYTING2:th.26">Th26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c7">c</font> as    <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>by </b><i><a class="txt" href="heyting2.html#E3:42"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="heyting2.html#E1:42_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="heyting2.html#T26" target="_self" title="HEYTING2:th.26">Th26</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:42_1"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c9">b</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="c5">L</font>
 <b>and </b><br/><a NAME="E6:42_1"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:42"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="heyting2.html#E2:42_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="heyting2.html#T19" target="_self" title="HEYTING2:th.19">Th19</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:42_1"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font>
 <b>and </b><br/><a NAME="E9:42_1"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c5">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:42_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="substlat.html#T10" title="SUBSTLAT:th.10">SUBSTLAT:10</a></i>;<br/><b>take </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">d</font>;<br/><b>thus </b><a NAME="E10:42_1"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K4" title="HEYTING2:func.4">pseudo_compl</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:42"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="heyting2.html#E9:42_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="heyting2.html#D4" target="_self" title="HEYTING2:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E11:42_1"/>
<font color="Maroon" title="c11">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E6:42_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="heyting2.html#E8:42_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<span class="p1">(<span class="default"><a href="heyting2.html#K8" title="HEYTING2:func.8">Atom</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="heyting2.html#K4" title="HEYTING2:func.4">pseudo_compl</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E40"><i><font color="Green" title="E31">Lm8</font></i></a></i>;<br/>


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