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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/>









<b>assume </b><a NAME="E1:25"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="substlat.html#K4">^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
(  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:25_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="heyting1.html#D2">HEYTING1:def 2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> );<br/><div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:25_1_1_1"/>
not <font color="Maroon">s</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:25_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:25_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">t1</font></span>)</span> where <font color="Olive">s1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>, <font color="Olive">t1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : ( <font color="Olive">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s1</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">t1</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="heyting2.html#E2:25"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="substlat.html#D3">SUBSTLAT:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:25_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E7:25_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:25_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">V</font></span><a href="heyting1.html#K1">]</a></span> <b>by </b><i><a class="txt" href="heyting2.html#E2:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="heyting1.html#D2">HEYTING1:def 2</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E10:25_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="txt" href="heyting2.html#E2:25"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">V</font></span><a href="heyting1.html#K1">]</a></span><b> such that </b><br/><a NAME="E5:25_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font>]
 <b>from </b><i><a class="ref" href="fraenkel.html#S27">FRAENKEL:sch 27</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/><b>reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/><a NAME="E7:25_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">AA</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i><b>then reconsider </b><font color="Maroon">UKA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1">]</a></span> : verum </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a></i>;<br/><b>set </b><font color="Maroon">f</font> = <font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font>;<br/><a NAME="E9:25_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><b>consider </b><font color="Maroon">kk</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:25_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">kk</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">kk</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E12:25_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">kk</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/><a NAME="E13:25_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:25_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span>
 ;<br/>

<a NAME="E2:25_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UKA</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1">]</a></span><b> such that </b><br/><a NAME="E4:25_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as  <a href="finset_1.html#V1">finite</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<a NAME="E7:25_1_1_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">b'</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E8:25_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E14:25_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="heyting2.html#E2:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="heyting2.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> <b>by </b><i><a class="ref" href="heyting2.html#T2" target="_self">Th2</a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><a NAME="E16:25_1_1"/>
for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g1</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>assume </b><a NAME="E2:25_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:25_1_1_3"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span></span>)</span> &amp; <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">z</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font>;<br/>
<a NAME="E1:25_1_1_3_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:25_1_1_3_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:25_1_1_3_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font>
;<br/>

<a NAME="E4:25_1_1_3_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b1</font></span>)</span> where <font color="Olive">b1</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="heyting1.html#K1">[</a><span class="default"><font color="Maroon">A</font></span><a href="heyting1.html#K1">]</a></span> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:25"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E5:25_1_1_3_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">UKA</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:25_1_1_3_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_1_3_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E8:25_1_1_3_1"/>
<font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="heyting2.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_1_3"/>
not <font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:25_1_1"/><b>then </b>
<font color="Maroon">f'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f1</font> where <font color="Olive">f1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f1</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font> </span>}</span> 
 ;<br/><a NAME="E18:25_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">UKA</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><a NAME="E19:25_1_1"/>
 ex <font color="Olive">b</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">K</font> =  <a href="xboole_0.html#K1">{}</a>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b>
<font color="Maroon">K</font>
;<br/><b>thus </b><a NAME="E4:25_1_2"/>
( <font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K2" target="_self">-</a> <font color="Maroon">A</font> &amp; <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
