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

<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 <font color="Maroon" title="c2">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>] <b>means </b>for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds  $3 <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $1,<span class="p1">(<span class="default">$2 <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span>;<br/>
<a NAME="E1:9_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>  ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">z</font>]</span><br/><b>let </b><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">d</font>,<font color="Olive" title="b1">z</font>]</span><br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span> );<br/>
<a NAME="E1:9_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> implies  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>2</sub></font>[<font color="Maroon" title="c6">w</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<b>assume </b><a NAME="E1:9_1_1_1_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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>2</sub></font>[<font color="Maroon" title="c6">w</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">w</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>]</span><br/>

<b>thus </b><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">z</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E1:9_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> )</span><br/><b>assume </b><a NAME="E1:9_1_1_1_2"/>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span><br/><a NAME="E2:9_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> st <br/>( <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:9_1_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_1_2"/>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">z'</font> = <font color="Maroon" title="c6">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> <b>by </b><i><a class="txt" href="#E3:9_1_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">z'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">z'</font>]</span><br/>

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z'</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/>

<a NAME="E6:9_1_1_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:9_1_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1"/>
<font color="Maroon" title="c7">z'</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:9_1_1"/>
 ex <font color="Olive" title="b1">P</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"><font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">d</font>,<font color="Olive" title="b1">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">d</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:9_1_1"><i><font color="Green" title="E6">A1</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
