<?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> ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  ex <font color="Olive" title="b2">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Olive" title="b2">c</font> );<br/>
<a NAME="E1:18_1_1"/><i><font color="Green" title="E11">A1</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 <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</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>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_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> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>and </b><br/><a NAME="E2:18_1_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  <b>by </b><i><a class="txt" href="bspace.html#E1:18_1_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="bspace.html#E2:18_1_1_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c4">x</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c5">y</font>;<br/>
<b>take </b>
<font color="Maroon" title="c4">x</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c5">y</font>
;<br/>

<b>thus </b><a NAME="E5:18_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</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="c4">x</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c5">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green" title="E12">A4</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 <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</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">f</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#S1" title="BINOP_1:sch.1">BINOP_1:sch 1</a>(<a class="txt" href="bspace.html#E1:18_1_1"><i><font color="Green" title="E11">A1</font></i></a>);<br/></i>
<a NAME="E4:18_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <br/> for <font color="Olive" title="b2">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Olive" title="b2">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> ;<br/><b>let </b><font color="Maroon" title="c4">c</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> , <font color="Maroon" title="c6">c'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:18_1_1_2"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font>
 <b>and </b><br/><a NAME="E3:18_1_1_2"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font>
 <b>and </b><br/><a NAME="E4:18_1_1_2"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c6">c'</font>
 <b>by </b><i><a class="txt" href="bspace.html#E3:18_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:18_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="bspace.html#E2:18_1_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="bspace.html#E3:18_1_1_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="bspace.html#E4:18_1_1_2"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>thus </b><a NAME="E5:18_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <br/> for <font color="Olive" title="b2">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="bspace.html#K4" title="BSPACE:func.4">\*\</a> <font color="Olive" title="b2">c</font>
 <b>by </b><i><a class="txt" href="bspace.html#E4:18_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>


</div>
