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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">f2</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<i><font color="Green">E29</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><a NAME="E2:53_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:53_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c</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">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:53_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><a NAME="E5:53_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:53_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:53_2"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c</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">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">c</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">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:53_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:53_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T12" target="_self">Th12</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<i><font color="Green">E63</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><a NAME="E2:53_4"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:53_4"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c</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">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:53_4"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><a NAME="E5:53_4"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:53_4"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:53_4"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c</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">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:53_4"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:53_4"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T13" target="_self">Th13</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<i><font color="Green">E72</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><a NAME="E2:53_6"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E3:53_6"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c</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">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:53_6"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><a NAME="E5:53_6"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:53_6"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="rfunct_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:53_6"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</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">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:53_6"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:53_6"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c</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">f1</font> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T12" target="_self">Th12</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T14" target="_self">Th14</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:53"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K5">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
