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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> implies  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f</font> = (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>);<br/>
<a NAME="E2:60"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T62" title="FUNCT_4:th.62">FUNCT_4:62</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) ) &amp; (  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/><span class="kw">thus </span><a NAME="E1:60_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c8">g</font> = (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:60_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; (  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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:60_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="funct_4.html#T62" title="FUNCT_4:th.62">FUNCT_4:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> implies <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:60_1_1"/>
<font color="Maroon" title="c9">x</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E5:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:60_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E7:60_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E8:60_1_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E9:60_1_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:60_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E7:60_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E8:60_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">given </span><font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E3:60_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</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">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:60_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E6:60_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c1">a</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E7:60_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c2">b</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="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E8:60_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E6:60_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a></span>;<br/><a NAME="E9:60_1"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E7:60_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a></span>;<br/><a NAME="E10:60_1"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/><a NAME="E11:60_1"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/><a NAME="E12:60_1"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E10:60_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E13:60_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E11:60_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E14:60_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E4:60_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E12:60_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/><span class="kw">hence </span><a NAME="E15:60_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:60"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
