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

<b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_1_2"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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="funct_5.html#K2" target="_self">proj2</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="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <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="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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="funct_5.html#K2" target="_self">proj2</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="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </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="E1:19_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:19_1_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2" target="_self">proj2</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="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> holds <br/><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:19_1_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2" target="_self">proj2</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="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> holds <br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font> ) )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2" target="_self">proj2</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="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/><a NAME="E2:19_1_2_1_1"/><b>then </b>
( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:19_1_2_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="E4:19_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
