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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> holds  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">A</font></span><br/>



<b>reconsider </b><font color="Maroon" title="c3">theA</font> =  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #) as  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">AA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">theA</font> <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">BB</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">GAA</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font>;<br/>
<a NAME="E4:20"/><i><font color="Green" title="E7">A1</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">B</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> implies ( <font color="Maroon" title="c7">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">B</font> ) )</span><br/><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">B</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c7">B</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">theA</font> ;<br/><a NAME="E3:20_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c8">C</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><div><i><font color="Green" title="E10">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c9">o</font></span><br/><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c8">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c9">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>;<br/><a NAME="E3:20_1_1"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #)
 ;<br/><b>hence </b><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c7">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c9">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E2:20_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon" title="c7">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 <b>by </b><i><a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">B</font></span><br/><a NAME="E6:20_1"/><i><font color="Green" title="E11">A6</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">OAB</font> =  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">B</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c8">C</font></span><br/><a NAME="E1:20_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c2">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #)
 ;<br/><b>thus </b><font color="Maroon" title="c9">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">o</font> = 
<font color="Maroon" title="c10">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c7">B</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="msafree2.html#E4:20_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="msualg_2.html#D8" title="MSUALG_2:def.8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c10">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="txt" href="msafree2.html#E1:20_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="msafree2.html#E2:20_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="msualg_2.html#D8" title="MSUALG_2:def.8">MSUALG_2:def 8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:20_1"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">B</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E6:20_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">GAA</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="msafree2.html#E4:20"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c5">BB</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">U1</font> be    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">BB</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c8">U1</font> implies <font color="Maroon" title="c7">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c8">U1</font> )</span><br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon" title="c5">BB</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c8">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c8">U1</font></span><br/><a NAME="E2:20_2"/><i><font color="Green" title="E10">A11</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U1</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #)
 <b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">B</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">theA</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U1</font> implies ( <font color="Maroon" title="c9">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c9">B</font> ) )</span><br/><b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green" title="E11">A12</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c9">B</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c9">B</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font> ;<br/><a NAME="E3:20_2_1"/><i><font color="Green" title="E12">A13</font></i>: 
<font color="Maroon" title="c10">C</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2_1"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><div><i><font color="Green" title="E13">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c11">o</font></span><br/><a NAME="E1:20_2_1_1"/>
<font color="Maroon" title="c10">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c11">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_2_1"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_2_1_1"/><b>then </b><i><font color="Green" title="E14">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>;<br/><a NAME="E3:20_2_1_1"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">A</font>
 ;<br/><b>hence </b><a NAME="E4:20_2_1_1"/>
<font color="Maroon" title="c9">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c11">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E2:20_2_1_1"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_2_1"/>
<font color="Maroon" title="c9">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 <b>by </b><i><a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c9">B</font></span><br/><a NAME="E6:20_2_1"/><i><font color="Green" title="E14">A16</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2_1"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">OAB</font> =  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c9">B</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c10">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c10">C</font></span><br/><a NAME="E1:20_2_1_2"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c12">o</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_2_1"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_2_1_2"/><i><font color="Green" title="E16">A18</font></i>: 
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">A</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> #)
 ;<br/><b>thus </b><font color="Maroon" title="c11">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">o</font> = 
<font color="Maroon" title="c12">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c9">B</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">o</font></span>)</span>
<b>by </b><i><a class="txt" href="msafree2.html#E4:20_2_1"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="msafree2.html#E2:20_2_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="msualg_2.html#D8" title="MSUALG_2:def.8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c12">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c10">C</font>
<b>by </b><i><a class="txt" href="msafree2.html#E1:20_2_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="msualg_2.html#D8" title="MSUALG_2:def.8">MSUALG_2:def 8</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:20_2_1"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c3">theA</font>,<font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E6:20_2_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">U2</font> = <font color="Maroon" title="c8">U1</font> as    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c3">theA</font> <b>by </b><i><a class="txt" href="msafree2.html#E2:20_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/><a NAME="E5:20_2"/>
<font color="Maroon" title="c7">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c9">U2</font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>;<br/><b>hence </b><a NAME="E6:20_2"/>
<font color="Maroon" title="c7">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c8">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:20"/><b>then </b>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">AA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">BB</font>
 
<b>by </b><i><a class="txt" href="msafree2.html#E7:20"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E10:20"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">BB</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T22" title="MSUALG_2:th.22">MSUALG_2:22</a></i>;<br/>
<b>hence </b><a NAME="E11:20"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="msafree.html#D4" title="MSAFREE:def.4">MSAFREE:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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