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<b>let </b><font color="Maroon">S</font> be  non <a href="msualg_1.html#V2">void</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">E</font> be  non <a href="struct_0.html#V3">empty</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">E</font>;<br/>





<b>assume </b><a NAME="E1:69"/>
<font color="Maroon">A</font> is    <a href="algspec1.html#M4" target="_self">Algebra</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">ES</font> being  non <a href="msualg_1.html#V2">void</a>  <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is  <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">ES</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">ES</font> <b>by </b><i><a class="ref" href="algspec1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E5:69"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S</font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">ES</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:69"/><b>then </b><i><font color="Green">E27</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">ES</font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">ES</font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="instalg1.html#T13">INSTALG1:13</a></i>;<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/>

<a NAME="E7:69"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">ES</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T9" target="_self">Th9</a>, <a class="ref" href="instalg1.html#T11">INSTALG1:11</a></i>;<br/>
<a NAME="E8:69"/><i><font color="Green">E30</font></i>: 
( the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">ES</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">ES</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">ES</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">ES</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:69"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ES</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:69"/>
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">ES</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E11:69"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">ES</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="algspec1.html#T11" target="_self">Th11</a>, <a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E12:69"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">ES</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E13:69"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">ES</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="algspec1.html#T11" target="_self">Th11</a>, <a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E14:69"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">ES</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">ES</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T11" target="_self">Th11</a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E15:69"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="algspec1.html#T15" target="_self">Th15</a>, <a class="ref" href="algspec1.html#T16" target="_self">Th16</a></i>;<br/>


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