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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>, <font color="Maroon">E</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">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="msafree1.html#V1">disjoint_valued</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">E</font>
 ;<br/>

<b>then 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">E</font> ;<br/>
<a NAME="E3:74"/><i><font color="Green">E26</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green">E27</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a> &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">S</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T9" target="_self">Th9</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:74"/><i><font color="Green">E28</font></i>: 
(  <a href="relset_1.html#K4">dom</a> 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#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">o</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>;<br/><a NAME="E5:74_1"/>
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 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> &amp;  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">e</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 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">e</font> )
 ;<br/><a NAME="E6:74_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">e</font>,<font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E7:74_1"/><b>then </b>
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">e</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then </b> <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</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">E</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="algspec1.html#T10" target="_self">Th10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><a NAME="E9:74_1"/><b>then </b>
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">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span> <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">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E10:74_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="algspec1.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:74"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="algspec1.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E8:74"/><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K4">dom</a> 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#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:74_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">o</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> <a href="finseq_2.html#K3">*</a>  ;<br/><b>reconsider </b><font color="Maroon">q</font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  ;<br/><a NAME="E6:74_2"/>
(  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 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> &amp;  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">e</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 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">e</font> &amp;  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">e</font>,<font color="Maroon">B</font> is   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> )
 ;<br/><a NAME="E7:74_2"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">e</font>,<font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b> <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</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#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/><b>then </b> <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> = 
<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> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
;<br/><a NAME="E10:74_2"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</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#NK3" target="_self">*</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/><a NAME="E11:74_2"/><i><font color="Green">E33</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 ;<br/><a NAME="E12:74_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="algspec1.html#T15" target="_self">Th15</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>hence </b><a NAME="E13:74_2"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="algspec1.html#T15" target="_self">Th15</a>, <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, , <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:74"/>
 <a href="msualg_1.html#G1">ManySortedSign</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font>,the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font>,the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font>,the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> #)
 <b>by </b><i><a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="algspec1.html#T18" target="_self">Th18</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
