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

<b>let </b><font color="Maroon">A</font> be  <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be  <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">B1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/>





<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span>;<br/>
<a NAME="E1:36"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:36"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:36"/>
<font color="Maroon">f</font> is <a href="altcat_1.html#V7" target="_self">compositional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D11" target="_self">ALTCAT_1:def 11</a><br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
 ex <font color="Olive">h</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">h</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">h</font> )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36"/>
 <a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A1</font>,<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">A1</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="altcat_1.html#T3" target="_self">Th3</a></i>;<br/>


</div>
