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

<b>let </b><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <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>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> )
 ;<br/>

<a NAME="E2:12"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="substlat.html#T8" title="SUBSTLAT:th.8">SUBSTLAT:8</a></i>;<br/>
<b>thus </b><a NAME="E3:12"/>
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c5">a</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="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E2:12"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="substlat.html#T15" title="SUBSTLAT:th.15">SUBSTLAT:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:12_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c8">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<a NAME="E6:12_1"/>
( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">b1'</font> = <font color="Maroon" title="c8">b1</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="txt" href="heyting2.html#E3:12_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<a NAME="E8:12_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c10">b1'</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c9">b'</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="partfun1.html#T135" title="PARTFUN1:th.135">PARTFUN1:135</a></i>;<br/>
<a NAME="E9:12_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E10:12_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="heyting2.html#E5:12_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="heyting2.html#E8:12_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="substlat.html#T16" title="SUBSTLAT:th.16">SUBSTLAT:16</a></i>;<br/>
<a NAME="E11:12_1"/>
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="heyting2.html#E3:12_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="heyting2.html#E10:12_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="substlat.html#T6" title="SUBSTLAT:th.6">SUBSTLAT:6</a></i>;<br/>


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

<b>thus </b><a NAME="E4:12"/>
<font color="Maroon" title="c3">A</font> <a href="heyting1.html#R1" title="HEYTING1:pred.1">c=</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E4:12_2"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="substlat.html#T11" title="SUBSTLAT:th.11">SUBSTLAT:11</a></i>;<br/>
<a NAME="E5:12_2"/><i><font color="Green" title="E14">A11</font></i>: 
<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="txt" href="heyting2.html#E1:12_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="heyting2.html#T2" target="_self" title="HEYTING2:th.2">Th2</a></i>;<br/>
<a NAME="E6:12_2"/>
( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c6">b</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="txt" href="heyting2.html#E1:12_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<a NAME="E8:12_2"/><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E9:12_2"/>
<font color="Maroon" title="c7">a'</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c8">b'</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="partfun1.html#T135" title="PARTFUN1:th.135">PARTFUN1:135</a></i>;<br/>
<a NAME="E10:12_2"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="heyting2.html#E8:12_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="substlat.html#T16" title="SUBSTLAT:th.16">SUBSTLAT:16</a></i>;<br/>
<a NAME="E11:12_2"/>
for <font color="Olive" title="b1">b</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</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" title="c9">b</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>assume </b><a NAME="E1:12_2_1"/><i><font color="Green" title="E17">A14</font></i>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">c</font>, <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="E3:12_2_1"/><i><font color="Green" title="E18">A15</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="ref" href="substlat.html#T15" title="SUBSTLAT:th.15">SUBSTLAT:15</a></i>;<br/>
<a NAME="E4:12_2_1"/>
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E3:12_2_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:12_2_1"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:12_2_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:12_2_1"/><b>then </b>
<font color="Maroon" title="c10">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="txt" href="heyting2.html#E4:12_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="substlat.html#T6" title="SUBSTLAT:th.6">SUBSTLAT:6</a></i>;<br/>
<b>hence </b><a NAME="E7:12_2_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:12_2_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="heyting2.html#E3:12_2_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="heyting2.html#E4"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:12_2"/>
<font color="Maroon" title="c5">a</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="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="heyting2.html#E5:12_2"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="heyting2.html#E10:12_2"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="substlat.html#T7" title="SUBSTLAT:th.7">SUBSTLAT:7</a></i>;<br/>


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


</div>
