<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <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="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>assume </b><a NAME="E1:12"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <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>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</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> )
 ;<br/>

<a NAME="E2:12"/><i><font color="Green">E10</font></i>: 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="heyting1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="substlat.html#T8">SUBSTLAT:8</a></i>;<br/>
<b>thus </b><a NAME="E3:12"/>
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="heyting1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:12_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</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>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E2:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="substlat.html#T15">SUBSTLAT:15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12"><i><font color="Green">E9</font></i></a>, <a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E6:12_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E8:12_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="partfun1.html#T135">PARTFUN1:135</a></i>;<br/>
<a NAME="E9:12_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E10:12_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="heyting2.html#E8:12_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="substlat.html#T16">SUBSTLAT:16</a></i>;<br/>
<a NAME="E11:12_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting2.html#E10:12_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="substlat.html#T6">SUBSTLAT:6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:12"/>
<font color="Maroon">c<sub>3</sub></font> <a href="heyting1.html#R1">c=</a>  <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:12_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E4:12_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3">mi</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="substlat.html#T11">SUBSTLAT:11</a></i>;<br/>
<a NAME="E5:12_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="heyting2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:12_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</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#E1:12_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E8:12_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E9:12_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="partfun1.html#T135">PARTFUN1:135</a></i>;<br/>
<a NAME="E10:12_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="heyting2.html#E8:12_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="substlat.html#T16">SUBSTLAT:16</a></i>;<br/>
<a NAME="E11:12_2"/>
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>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_2_1"/><i><font color="Green">E17</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> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_2_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="substlat.html#T15">SUBSTLAT:15</a></i>;<br/>
<a NAME="E4:12_2_1"/>
<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#E3:12_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:12_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:12_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E4:12_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="substlat.html#T6">SUBSTLAT:6</a></i>;<br/>
<b>hence </b><a NAME="E7:12_2_1"/>
<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="txt" href="heyting2.html#E1:12_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="heyting2.html#E4"><i><font color="Green">Lemma3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:12_2"/>
<font color="Maroon">c<sub>5</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>3</sub></font> <a href="substlat.html#K4">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:12_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="heyting2.html#E10:12_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="substlat.html#T7">SUBSTLAT:7</a></i>;<br/>


</div><b>end;</b></div>


</div>
