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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:15_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> 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>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) iff <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substlat.html#D2">SUBSTLAT:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</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>3</sub></font><b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> ) iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) ) )
 ;<br/><a NAME="E5:15_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E4:15_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:15_1"/>
 <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="ref" href="partfun1.html#T128">PARTFUN1:128</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="heyting3.html#E4:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="heyting3.html#E5:15_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E8:15_1"/>
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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <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>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> ) iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting3.html#E1:15_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting3.html#E4:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="substlat.html#T6">SUBSTLAT:6</a></i>;<br/><a NAME="E9:15_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) iff <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting3.html#E4:15_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E10:15_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="substlat.html#D2">SUBSTLAT:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:15"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E4:15"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <font color="Olive">b<sub>2</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="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) iff <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substlat.html#D2">SUBSTLAT:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:15"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; ( 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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <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>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> ) iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="heyting3.html#E6:15"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E8:15"/><i><font color="Green">E14</font></i>: 
 <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="substlat.html#T8">SUBSTLAT:8</a></i>;<br/>
<a NAME="E9:15"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting3.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="substlat.html#T6">SUBSTLAT:6</a></i>;<br/>
<b>hence </b><a NAME="E10:15"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting3.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting3.html#E8:15"><i><font color="Green">E14</font></i></a>, <a class="ref" href="substlat.html#T7">SUBSTLAT:7</a></i>;<br/>


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