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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> iff <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">d</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K2">"</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="cqc_lang.html#T44">CQC_LANG:44</a></i>
<br/>.= 
<font color="Maroon">d</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">d</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T44">CQC_LANG:44</a></i>
;<br/><b>hence </b><a NAME="E3:14_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">f</font> = <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:14_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:14_1_2_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:14_1_2_1"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_2_1_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_2_1_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E3:14_1_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:14_1_2_1_1_3"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_2_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E3:14_1_2_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E4:14_1_2_1_1_3"/>
verum
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><i><font color="Green">E56</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>
;<br/><a NAME="E4:14_1_2"/>
for <font color="Olive">y</font>, <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:14_1_2_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>and </b><br/><a NAME="E2:14_1_2_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:14_1_2_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_3_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_2_3_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:14_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:14_1_2_3_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_2_3_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:14_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:14_1_2_3_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:14_1_2_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> implies ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2_3_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>and </b><br/><a NAME="E2:14_1_2_3_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:14_1_2_3_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_2_3_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_2_1_1"><b>suppose </b></a><a NAME="E1:14_1_2_3_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_3_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:14_1_2_3_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:14_1_2_3_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_3_2_1_2"><b>suppose </b></a><a NAME="E1:14_1_2_3_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:14_1_2_3_2_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:14_1_2_3_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:14_1_2_3_2_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1_2_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:14_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
