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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">X</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>, <font color="Maroon">Y</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:38"/><i><font color="Green">E42</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">SA</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">SB</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">SA</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; <font color="Maroon">SB</font> is <a href="msualg_2.html#V3">opers_closed</a> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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="E1:38_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/><a NAME="E3:38_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">SA</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o</font> &amp; <font color="Maroon">SB</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o</font> )
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><i><font color="Green">E49</font></i>: the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">SA</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>
<br/>.= 
<font color="Maroon">o</font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">SA</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</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"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">SA</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
;<br/><i><font color="Green">E50</font></i>: the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">SB</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>
<br/>.= 
<font color="Maroon">o</font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">SB</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</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"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">SB</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
;<br/><a NAME="E6:38_1"/>
( <font color="Maroon">SA</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">SB</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:38_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SA</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SB</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#T3">MSUALG_2:3</a></i>;<br/><a NAME="E8:38_1"/>
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/><a NAME="E9:38_1"/><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>,<span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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>,<span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:38_1"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SA</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SB</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3"><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="E1:38_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SA</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">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> as    <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="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">x</font> as    <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="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<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="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="instalg1.html#T4">INSTALG1:4</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"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="instalg1.html#T4">INSTALG1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:38_1"/>
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">x</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">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
 <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> #)
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
