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





<b>assume </b><a NAME="E1:3"/><i><font color="Green">E37</font></i>: 
 <a href="extens_1.html#K2">rngs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:3"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I</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="E3:3"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">B</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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="E1:3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:3_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<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">f</font> <a href="relat_1.html#K7">|</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">B</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="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E3:3_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span> <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="E4:3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="extens_1.html#K2">rngs</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</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="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E5:3_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E6:3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</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> &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K8">|</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</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> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K7">|</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="relat_1.html#T112">RELAT_1:112</a>, <a class="ref" href="relat_1.html#T125">RELAT_1:125</a></i>;<br/><a NAME="E7:3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="relat_1.html#K7">|</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <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">f</font> <a href="relat_1.html#K7">|</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="monoid_1.html#T2">MONOID_1:2</a></i>;<br/><b>hence </b><a NAME="E8:3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">B</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">B</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="catalg_1.html#K1" target="_self">-MSF</a> <font color="Maroon">I</font>,<font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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