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

<b>let </b><font color="Maroon">I1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I1</font>,<font color="Maroon">I2</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2</font>;<br/>









<b>assume </b><a NAME="E1:8"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>;<br/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 ;<br/><b>hence </b><a NAME="E2:8_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E3:8_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <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="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E4:8_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 ;<br/><a NAME="E2:8_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E3:8_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:8_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>,<font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_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="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:8_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 ;<br/>

<a NAME="E2:8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:8_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="relset_1.html#M2">M6</a>(<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>)
 ;<br/>


</div><b>end;</b></div><a NAME="E5:8_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <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="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E6:8_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_2.html#K3">**</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <font color="Maroon">B</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_2.html#K3">**</a> <font color="Maroon">M</font> <a href="pboole.html#R6">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
