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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I0</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">A0</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I0</font>, <font color="Maroon">B0</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I0</font>;<br/>









<b>assume </b><a NAME="E1:10"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">A0</font> <a href="pboole.html#R6">=</a> <font color="Maroon">A</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font> &amp; <font color="Maroon">B0</font> <a href="pboole.html#R6">=</a> <font color="Maroon">B</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font> )
 ;<br/>

<a NAME="E2:10"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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="E1:10_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I0</font>
 ;<br/><a NAME="E2:10_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E3:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I0</font> <b>by </b><i>, <a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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="E1:10_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I0</font>
 ;<br/><a NAME="E2:10_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E3:10_2"/>
( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:10_2"/>
<font color="Maroon">G</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">A0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B0</font> <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/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:10"/>
<font color="Maroon">F</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">I0</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A0</font>,<font color="Maroon">B0</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


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