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<b>let </b><font color="Maroon">J</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">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <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">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:63"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 ;<br/>

<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E2:63"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</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">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<a NAME="E2:63_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:63_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:63_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font>,<font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E7:63"/>
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</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">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:63"/>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_3.html#K2">id</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>;<br/>


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