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<b>let </b><font color="Maroon">I</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">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green">E38</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">i'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E2:28"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<i><font color="Green">E43</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">A</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D20">PRALG_2:def 20</a></i>
<br/>.= 
 <a href="pralg_2.html#K2">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<a NAME="E4:28"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E45</font></i>: 
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:28"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</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/>
<font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="pralg_2.html#D6">PRALG_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="pralg_1.html#K15">..</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>
;<br/>
<b>then </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</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="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</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="pralg_2.html#T14">PRALG_2:14</a></i>
;<br/>
<b>hence </b><a NAME="E9:28"/>
<font color="Maroon">g</font> <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_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 ;<br/>


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