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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>, <font color="Maroon" title="c4">v</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="c5">a</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E39">A1</font></i>: 
for <font color="Olive" title="b1">b</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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> holds <br/><font color="Olive" title="b1">b</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c5">a</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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="c5">a</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">v</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">u'</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c7">v'</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c8">Aa</font> = <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="c5">a</font> as    <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> <b>by </b><i><a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></i>;<br/>
<div><i><font color="Green" title="E41">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font>
 ;<br/><a NAME="E2:46_1"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u'</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">c'</font> = <font color="Maroon" title="c9">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="ref" href="setwiseo.html#T14" title="SETWISEO:th.14">SETWISEO:14</a></i>;<br/><a NAME="E4:46_1"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Aa</font>
 <b>by </b><i><a class="ref" href="heyting2.html#T28" target="_self" title="HEYTING2:th.28">Th28</a></i>;<br/><a NAME="E5:46_1"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c10">c'</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="heyting2.html#E1:46_1"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/><a NAME="E6:46_1"/>
<font color="Maroon" title="c9">c</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="heyting2.html#E2:46_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="heyting2.html#T2" target="_self" title="HEYTING2:th.2">Th2</a></i>;<br/><a NAME="E7:46_1"/><b>then </b><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/><a NAME="E8:46_1"/>
<font color="Maroon" title="c9">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a</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">s1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">t1</font></span>)</span> where <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">t1</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">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u'</font> &amp; <font color="Olive" title="b2">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Aa</font> &amp; <font color="Olive" title="b1">s1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b2">t1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="heyting2.html#E4:46_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="heyting2.html#E5:46_1"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/><a NAME="E9:46_1"/><b>then </b>
<font color="Maroon" title="c9">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u'</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">Aa</font>
 <b>by </b><i><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="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:46_1"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a</font>
 <b>and </b><br/><a NAME="E12:46_1"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c11">b</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"><font color="Maroon" title="c6">u'</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">Aa</font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E7:46_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="substlat.html#T10" title="SUBSTLAT:th.10">SUBSTLAT:10</a></i>;<br/><a NAME="E13:46_1"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</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> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">u</font>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E12:46_1"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></i>;<br/><a NAME="E14:46_1"/><b>then </b>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<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="c5">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:46_1"/><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font>
 <b>and </b><br/><a NAME="E17:46_1"/><i><font color="Green" title="E50">A12</font></i>: 
<font color="Maroon" title="c12">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E2:46"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="heyting2.html#E45"><i><font color="Green" title="E36">Lm9</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c13">e</font> = <font color="Maroon" title="c12">d</font>;<br/><b>thus </b><a NAME="E18:46_1"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E16:46_1"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/><b>thus </b><a NAME="E19:46_1"/>
<font color="Maroon" title="c13">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E11:46_1"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="heyting2.html#E17:46_1"><i><font color="Green" title="E50">A12</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E42">A13</font></i>: 
<font color="Maroon" title="c9">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="c5">a</font>
 ;<br/><a NAME="E2:46_2"/><b>then </b>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="heyting2.html#T26" target="_self" title="HEYTING2:th.26">Th26</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:46_2"/><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u'</font> <a href="heyting2.html#K3" title="HEYTING2:func.3">=&gt;&gt;</a> <font color="Maroon" title="c7">v'</font>
 <b>and </b><br/><a NAME="E5:46_2"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E4:46"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="heyting2.html#T29" target="_self" title="HEYTING2:th.29">Th29</a></i>;<br/><a NAME="E6:46_2"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Aa</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46_2"><i><font color="Green" title="E42">A13</font></i></a></i>;<br/><a NAME="E7:46_2"/><b>then </b>
<font color="Maroon" title="c9">c</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/><a NAME="E8:46_2"/><b>then </b>
<font color="Maroon" title="c10">b</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:46_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:46_2"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font>
 <b>and </b><br/><a NAME="E11:46_2"/><i><font color="Green" title="E46">A17</font></i>: 
<font color="Maroon" title="c11">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"><font color="Maroon" title="c6">u'</font> <a href="heyting2.html#K3" title="HEYTING2:func.3">=&gt;&gt;</a> <font color="Maroon" title="c7">v'</font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E4:46_2"><i><font color="Green" title="E43">A14</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="c12">e</font> = <font color="Maroon" title="c11">d</font>;<br/><b>thus </b><a NAME="E12:46_2"/>
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K5" title="HEYTING2:func.5">StrongImpl</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E11:46_2"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="heyting2.html#D5" target="_self" title="HEYTING2:def.5">Def5</a></i>;<br/><b>thus </b><a NAME="E13:46_2"/>
<font color="Maroon" title="c12">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:46_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="heyting2.html#E10:46_2"><i><font color="Green" title="E45">A16</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="E6:46"/>
<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="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="heyting2.html#K5" title="HEYTING2:func.5">StrongImpl</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E40"><i><font color="Green" title="E31">Lm8</font></i></a></i>;<br/>


</div>
