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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>



<a NAME="E1:87"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" target="_self" title="FUNCT_3:def.5">Def5</a>, <a class="ref" href="funct_3.html#D6" target="_self" title="FUNCT_3:def.6">Def6</a></span>;<br/>
<a NAME="E2:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T70" target="_self" title="FUNCT_3:th.70">Th70</a></span>;<br/>
<a NAME="E3:87"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:87_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E2:87_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="funct_3.html#D8" target="_self" title="FUNCT_3:def.8">Def8</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="funct_3.html#D5" target="_self" title="FUNCT_3:def.5">Def5</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="funct_3.html#D6" target="_self" title="FUNCT_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:87"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:87"/>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="funct_3.html#T6" target="_self" title="FUNCT_3:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
