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

<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</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>let </b><font color="Maroon" title="c3">A</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="E1:20"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">C</font>] <b>means </b>for <font color="Olive" title="b1">g</font> being   <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>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> not $1 <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b1">g</font>;<br/>
<a NAME="E2:20"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">xx</font> where <font color="Olive" title="b1">xx</font> is    <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">xx</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">C</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S10" title="FRAENKEL:sch.10">FRAENKEL:sch 10</a>();<br/></i>
<a NAME="E3:20"/><b>then </b>
 <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <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> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="heyting2.html#T11" target="_self" title="HEYTING2:th.11">Th11</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T122" title="PARTFUN1:th.122">PARTFUN1:122</a></i>;<br/>
<a NAME="E5:20"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:20_1"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T122" title="PARTFUN1:th.122">PARTFUN1:122</a></i>;<br/>
<a NAME="E3:20_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="heyting2.html#T11" target="_self" title="HEYTING2:th.11">Th11</a></i>;<br/>
<a NAME="E4:20_1"/>
for <font color="Olive" title="b1">g</font> being   <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>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/> not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b1">g</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:20"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E3:20_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:20"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
 <a href="heyting2.html#K2" title="HEYTING2:func.2">-</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="heyting2.html#E4:20"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
