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

<b>set </b><font color="Maroon" title="c10">Y</font> = <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>;<br/>
<b>set </b><font color="Maroon" title="c11">X</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>;<br/>
<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> ] <b>means </b>for <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:7_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">I</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:7_1_1"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">rj</font> = <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pralg_2.html#E1:7_1_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">pj</font> = <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">I</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="pralg_2.html#E1:7_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="pralg_2.html#E2:7_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<a NAME="E5:7_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:7_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">pj</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">rj</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E7:7_1_1"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>;<br/>
<a NAME="E8:7_1_1"/><i><font color="Green" title="E8">A5</font></i>: 
for <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c15">h1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:7_1_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c15">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> holds <br/><font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E7:7_1_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c16">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">y</font>] )</span><br/>

<a NAME="E4:7_1_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:7_1_1_1"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">pj</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">rj</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">pj</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">rj</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:7_1_1_1"/><b>then </b><i><font color="Green" title="E11">A8</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>;<br/>
<a NAME="E7:7_1_1_1"/><i><font color="Green" title="E12">A9</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:7_1_1_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E8:7_1_1_1"/><i><font color="Green" title="E13">A10</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:7_1_1_1"/><i><font color="Green" title="E14">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">pj</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:7_1_1_1"/><b>then </b><i><font color="Green" title="E15">A12</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:7_1_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="pralg_2.html#E8:7_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E11:7_1_1_1"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<a NAME="E12:7_1_1_1"/><b>then </b><i><font color="Green" title="E17">A14</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E13:7_1_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</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="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_1_1"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/>

<a NAME="E2:7_1_1_1_1"/><b>then </b><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:7_1_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="pralg_2.html#E10:7_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/>
<a NAME="E3:7_1_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E4:7_1_1_1_1"/><b>then </b><i><font color="Green" title="E20">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E5:7_1_1_1_1"/>
<font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:7_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="pralg_2.html#E2:7_1_1_1_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="pralg_2.html#E4:7_1_1_1_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:7_1_1_1_1"/><i><font color="Green" title="E21">A18</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> &amp; <font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="pralg_2.html#E10:7_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c18">a</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E10:7_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:7_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="pralg_2.html#E8:7_1_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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><b>end;</b></div>
<a NAME="E14:7_1_1_1"/><b>then </b>
 <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E15:7_1_1_1"/><b>then </b><i><font color="Green" title="E18">A19</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:7_1_1_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="pralg_2.html#E7:7_1_1_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:7_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E10:7_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<a NAME="E17:7_1_1_1"/><b>then </b><i><font color="Green" title="E19">A20</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E18:7_1_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</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="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_1_2"/><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/>

<a NAME="E2:7_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A22</font></i>: 
<font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:7_1_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="pralg_2.html#E10:7_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/>
<a NAME="E3:7_1_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E4:7_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A23</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E5:7_1_1_1_2"/>
<font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">pj</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:7_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:7_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="pralg_2.html#E2:7_1_1_1_2"><i><font color="Green" title="E21">A22</font></i></a>, <a class="txt" href="pralg_2.html#E4:7_1_1_1_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:7_1_1_1_2"/><i><font color="Green" title="E23">A24</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> &amp; <font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">pj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> &amp; <font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">a</font>,<font color="Maroon" title="c19">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="pralg_2.html#E10:7_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pralg_2.html#E11:7_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="pralg_2.html#E12:7_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="pralg_2.html#E1:7_1_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c19">b</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E8:7_1_1_1_2"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E10:7_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:7_1_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="txt" href="pralg_2.html#E8:7_1_1_1_2"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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><b>end;</b></div>
<a NAME="E19:7_1_1_1"/><b>then </b>
 <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">pj</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E17:7_1_1_1"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E20:7_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:7_1_1_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="pralg_2.html#E7:7_1_1_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:7_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E7:7_1_1_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="pralg_2.html#E15:7_1_1_1"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:7_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c15">h1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/>
<b>hence </b><a NAME="E23:7_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="ref" href="pralg_2.html#T9" target="_self" title="PRALG_2:th.9">Th9</a></i>; <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="c14">x</font>,<font color="Maroon" title="c16">y</font>]</span><br/>

<b>let </b><font color="Maroon" title="c17">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> implies <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>assume </b><a NAME="E24:7_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>hence </b><a NAME="E25:7_1_1_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c17">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:7_1_1_1"><i><font color="Green" title="E9">A6</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><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font><b> such that </b><br/><a NAME="E10:7_1_1"/><i><font color="Green" title="E9">A25</font></i>: 
for <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="pralg_2.html#E8:7_1_1"><i><font color="Green" title="E8">A5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c14">F</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">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> holds <br/><font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>thus </b><a NAME="E11:7_1_1"/>
for <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K9" title="PBOOLE:func.9">[|</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="pboole.html#K9" title="PBOOLE:func.9">|]</a></span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> holds <br/><font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E10:7_1_1"><i><font color="Green" title="E9">A25</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>
