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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">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font>, <font color="Maroon">G</font> be   <a href="pboole.html#NM2">ManySortedFunction</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>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K1">doms</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:8"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">FG</font> = <font color="Maroon">F</font> <a href="pboole.html#K12">**</a> <font color="Maroon">G</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="mssubfam.html#T15">MSSUBFAM:15</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">FG</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>
 
;<br/>
<div><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">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>reconsider </b><font color="Maroon">fg</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K12">**</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E5:8_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K1">doms</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="mssubfam.html#T14">MSSUBFAM:14</a></i>;<br/><a NAME="E6:8_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/><a NAME="E7:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</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="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">fg</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K12">**</a> <font color="Maroon">G</font></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K12">**</a> <font color="Maroon">G</font></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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