<?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">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:87"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Olive">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> );<br/>
<a NAME="E2:87"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:87_1"/><b>then </b>
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E5:87_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:87"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="ref" href="funct_6.html#T13" target="_self">Th13</a>);<br/></i>
<a NAME="E5:87"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/>(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Olive">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Olive">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:87_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:87_2"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font></span>)</span> ) &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:87_2_1"/>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i><a class="txt" href="funct_6.html#E3:87_2_1"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E5:87_2_1"/><b>then </b>
 ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:87_2_1"/>
<font color="Maroon">a</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:87_2"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp; <font color="Maroon">g</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:87_2_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:87_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="funct_6.html#E4:87_2_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="funct_6.html#E8:87_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:87_2"/><b>then </b>
(  <a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_6.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E7:87_2"/><b>then </b>
 <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:87_2"/><i><font color="Green">E58</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E4:87_2"><i><font color="Green">E55</font></i></a>, , </i>;<br/>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:87_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:87_2"/><i><font color="Green">E59</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"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="funct_6.html#E8:87_2"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E13:87_2"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="txt" href="funct_6.html#E8:87_2"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_2_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:87_2_3"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:87_2_3"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E5:87_2_3"/>
( <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="txt" href="funct_6.html#E8:87_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="funct_6.html#E11:87_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="funct_6.html#E13:87_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="funct_6.html#E1:87_2_3"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/><b>hence </b><a NAME="E6:87_2_3"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="funct_6.html#E1:87_2_3"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:87_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="funct_6.html#E11:87_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E6:87"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:87_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">gx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:87_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gx</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">gx</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">gy</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:87_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gy</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">gy</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:87_3"/><i><font color="Green">E92</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gx</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gy</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">gx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">gy</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font></span>)</span> ) &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , , , <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_3_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:87_3_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">gx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">gy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">hx</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:87_3_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">gx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hx</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">hx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">hx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">hy</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:87_3_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">gy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hy</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">hy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">hy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E7:87_3_1"/><i><font color="Green">E116</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">gx</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">gy</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">gx</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">gx</font> &amp;  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">gy</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">gy</font> &amp; <font color="Maroon">gx</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">gx</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gx</font> &amp; <font color="Maroon">gy</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">gy</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gy</font> )
 <b>by </b><i>, , , , , <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_3_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">fx</font> = <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">gx</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funct_6.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E3:87_3_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gx</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_6.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E4:87_3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">gx</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">gx</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">gy</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">gy</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="funct_6.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/><b>hence </b><a NAME="E5:87_3_1_1"/>
<font color="Maroon">hx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:87_3_1"/>
<font color="Maroon">gx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:87_3"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:87"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:87"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">f</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_4"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:87_4"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:87_4"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">g</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E7:87_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:87"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:87"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/><a NAME="E2:87_5"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">j</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>hence </b><a NAME="E3:87_5"/>
<font color="Maroon">z</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E13:87"/><b>then </b><i><font color="Green">E134</font></i>: 
(  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_6"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:87_6"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:87_6"/><b>then </b><i><font color="Green">E136</font></i>: 
 ex <font color="Olive">j</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">j</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<a NAME="E15:87"/><b>then </b><i><font color="Green">E139</font></i>: 
( <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">h</font> &amp;  <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E16:87"/><b>then </b><i><font color="Green">E141</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="funct_6.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_7"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/><a NAME="E3:87_7"/><i><font color="Green">E144</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E4:87_7"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_7_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:87_7_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:87_7_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">z</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">j</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , , , , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>
;<br/>
<b>hence </b><a NAME="E8:87_7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E5:87_7"/><b>then </b>
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:87_7"/>
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:87"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" target="_self">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E19:87"/><b>then </b><i><font color="Green">E149</font></i>: 
 ex <font color="Olive">g'</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g'</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Olive">g'</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E20:87"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:87"/><b>then </b><i><font color="Green">E150</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_6.html#T12" target="_self">Th12</a>, , <a class="ref" href="relat_1.html#T64">RELAT_1:64</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:87"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K6">uncurry'</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="funct_6.html#K4" target="_self">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <span class="p2"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_5.html#T33">FUNCT_5:33</a></i>;<br/>
<a NAME="E23:87"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_5.html#K6">uncurry'</a> <font color="Maroon">h</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K4" target="_self">meet</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K6">uncurry'</a> <font color="Maroon">h</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K4" target="_self">meet</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <span class="p4"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font> &amp;  <a href="funct_5.html#K6">uncurry'</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="funct_4.html#K2">~</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span> &amp;  <a href="funct_5.html#K3">curry</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K5">curry'</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span> &amp;  <a href="funct_5.html#K6">uncurry'</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K4">uncurry</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_5.html#T56">FUNCT_5:56</a>, <a class="ref" href="funct_5.html#D5">FUNCT_5:def 5</a>, <a class="ref" href="funct_5.html#D6">FUNCT_5:def 6</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<a NAME="E24:87"/><b>then </b>
<span class="p1"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" target="_self">&lt;:</a><span class="default"><font color="Maroon">h</font></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span></span><a href="funct_6.html#K6" target="_self">:&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#T55">FUNCT_5:55</a></i>;<br/>
<b>hence </b><a NAME="E25:87"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
