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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> 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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <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>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <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><a NAME="E1:42"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:42"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:42"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</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/>
<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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</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>6</sub></font>
 ;<br/><a NAME="E2:42_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E3:42"><i><font color="Green">E36</font></i></a>, <a class="ref" href="heyting2.html#T26" target="_self">Th26</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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="txt" href="heyting2.html#E3:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="heyting2.html#E1:42_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="heyting2.html#T26" target="_self">Th26</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:42_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:42_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="heyting2.html#E2:42_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="heyting2.html#T19" target="_self">Th19</a></i>;<br/><b>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="E8:42_1"/><i><font color="Green">E41</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>and </b><br/><a NAME="E9:42_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:42_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="substlat.html#T10">SUBSTLAT:10</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E10:42_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting2.html#K4" target="_self">pseudo_compl</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>3</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="heyting2.html#E9:42_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="heyting2.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E11:42_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E6:42_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="heyting2.html#E8:42_1"><i><font color="Green">E41</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="E5:42"/>
<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>6</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="heyting2.html#K4" target="_self">pseudo_compl</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>3</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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