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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<div><i><font color="Green">E41</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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:46_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E4:46_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="heyting2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:46_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="heyting2.html#E1:46_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E6:46_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="heyting2.html#E2:46_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="heyting2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:46_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><a NAME="E8:46_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="heyting2.html#E4:46_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="heyting2.html#E5:46_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E9:46_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="substlat.html#D3">SUBSTLAT:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:46_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E12:46_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E7:46_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="substlat.html#T10">SUBSTLAT:10</a></i>;<br/><a NAME="E13:46_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E12:46_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/><a NAME="E14:46_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:46_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E17:46_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E2:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="heyting2.html#E45"><i><font color="Green">Lemma36</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E18:46_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E16:46_1"><i><font color="Green">E49</font></i></a></i>;<br/><b>thus </b><a NAME="E19:46_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E11:46_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="heyting2.html#E17:46_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K8" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:46_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="heyting2.html#T26" target="_self">Th26</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:46_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="heyting2.html#K3" target="_self">=&gt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:46_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E4:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="heyting2.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:46_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:46_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:46_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="heyting2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E8:46_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:46_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:46_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E11:46_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="heyting2.html#K3" target="_self">=&gt;&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E4:46_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="substlat.html#T10">SUBSTLAT:10</a></i>;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E12:46_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K5" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E11:46_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="heyting2.html#D5" target="_self">Def5</a></i>;<br/><b>thus </b><a NAME="E13:46_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:46_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="heyting2.html#E10:46_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T1">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" target="_self">Atom</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="heyting2.html#K5" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E40"><i><font color="Green">Lemma31</font></i></a></i>;<br/>


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