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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b2">m1</font> );<br/>
<a NAME="E1:37_3_1"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_3_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">m2</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">m1</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> ;<br/>
<b>assume </b><a NAME="E3:37_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>]
 ;<br/>

<a NAME="E4:37_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c6">m1</font>
 
;<br/>
<b>hence </b><a NAME="E5:37_3_1_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_3_1"/><i><font color="Green" title="E25">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z1</font>, <font color="Olive" title="b4">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">z2</font>] holds <br/><font color="Olive" title="b3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">z2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_3_1_2"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> ;<br/>
<b>assume </b><a NAME="E3:37_3_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">z2</font>] )
 ;<br/>

<a NAME="E4:37_3_1_2"/><b>then </b>
( <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">m1</font> )
 
;<br/>
<b>hence </b><a NAME="E5:37_3_1_2"/>
<font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">h</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:37_3_1"/><i><font color="Green" title="E26">A7</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) )
 <b>and </b><br/><a NAME="E5:37_3_1"/><i><font color="Green" title="E27">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c2">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S5" title="BINOP_1:sch.5">BINOP_1:sch 5</a>(<a class="txt" href="ens_1.html#E1:37_3_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="ens_1.html#E2:37_3_1"><i><font color="Green" title="E25">A6</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">h</font>
;<br/>

<b>thus </b><a NAME="E6:37_3_1"/><i><font color="Green" title="E28">A9</font></i>: 
for <font color="Olive" title="b1">m2</font>, <font color="Olive" title="b2">m1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m2</font>,<font color="Olive" title="b2">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> iff  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m2</font>, <font color="Maroon" title="c4">m1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>;<br/>



<b>thus </b><a NAME="E1:37_3_1_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> implies  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_3_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font>
 ;<br/>

<a NAME="E2:37_3_1_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font>,<font color="Olive" title="b1">z</font>]
 
<b>by </b><i><a class="txt" href="ens_1.html#E4:37_3_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:37_3_1_3_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:37_3_1_3"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font>
 ;<br/>

<a NAME="E3:37_3_1_3"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font>,<font color="Maroon" title="c3">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c4">m1</font>]
 
;<br/>
<b>hence </b><a NAME="E4:37_3_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E4:37_3_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">m2</font>, <font color="Maroon" title="c4">m1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E7:37_3_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m1</font>
 ;<br/>

<a NAME="E8:37_3_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E6:37_3_1"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>
<a NAME="E9:37_3_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font>,<font color="Maroon" title="c2">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font>]
 
<b>by </b><i><a class="txt" href="ens_1.html#E5:37_3_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:37_3_1"/>
<font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c4">m1</font>
 ;<br/>


</div>
