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<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font> holds   the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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/>



<span class="kw">reconsider </span><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#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A</font> #) as   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">AA</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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="E3:19"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> holds <br/>( <font color="Olive" title="b1">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp;  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c6">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp;  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</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="c6">B</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp;  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</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="c6">B</font>) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">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="E3:19_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">C</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><div><span class="lab"><font color="Green" title="E10">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c6">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Olive" title="b1">o</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c8">o</font></span><br/><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c7">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c8">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><a NAME="E2:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c8">o</font>,<a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(#  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A</font> #))</span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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="c8">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="c7">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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="c8">o</font>
 ;<br/><span class="kw">thus </span><a NAME="E3:19_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c8">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:19_1"/>
<font color="Maroon" title="c6">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</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="c6">B</font>)</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">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="c6">B</font>) as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c7">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> ;<br/><div><span class="lab"><font color="Green" title="E11">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c8">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c7">C</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c7">C</font></span><br/><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
( <font color="Maroon" title="c7">C</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c9">o</font> &amp;  <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#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A</font> #)) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c8">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">o</font> = 
<font color="Maroon" title="c9">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c6">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D8" onmouseover="rs('msualg_2/D8')" onmouseout="rh()">MSUALG_2:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><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>)</span>)</span> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="msualg_2.html#D7" onmouseover="rs('msualg_2/D7')" onmouseout="rh()">MSUALG_2:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c9">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c7">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="msualg_2.html#D7" onmouseover="rs('msualg_2/D7')" onmouseout="rh()">MSUALG_2:def 7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:19_1"/>
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</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="c7">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E9:19_1"/>
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#R8" title="PBOOLE:pred.8">=</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="c6">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="msualg_2.html#D8" onmouseover="rs('msualg_2/D8')" onmouseout="rh()">MSUALG_2:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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> as   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">BB</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E8">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U1</font> being    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c7">BB</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Olive" title="b1">U1</font> holds <br/><font color="Maroon" title="c6">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">U1</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c6">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/><div><span class="lab"><font color="Green" title="E9">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">theA</font>  st <font color="Olive" title="b1">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U1</font> holds <br/>( <font color="Olive" title="b1">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp;  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> (<font color="Maroon" title="c3">theA</font>,<font color="Olive" title="b1">B</font>) )</span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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/><span class="kw">assume </span><a NAME="E1:19_2_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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/><span class="kw">reconsider </span><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="E3:19_2_1"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon" title="c10">C</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><div><span class="lab"><font color="Green" title="E12">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c9">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Olive" title="b1">o</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E1:19_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E2:19_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K5" title="RELSET_1:func.5">|</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="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">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>
 ;<br/><span class="kw">thus </span><a NAME="E3:19_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1_1"><span class="lab"><font color="Green" title="E13">A13</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:19_2_1"/>
<font color="Maroon" title="c9">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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/><span class="kw">reconsider </span><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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c10">C</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> ;<br/><div><span class="lab"><font color="Green" title="E13">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c11">OAB</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c10">C</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E1:19_2_1_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
 <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#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A</font> #))
 ;<br/><a NAME="E2:19_2_1_2"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D8" onmouseover="rs('msualg_2/D8')" onmouseout="rh()">MSUALG_2:def 8</a></span>
<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#K5" title="RELSET_1:func.5">|</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="finseq_2.html#K6" title="FINSEQ_2:func.6">#</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_2_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E1:19_2_1_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="msualg_2.html#D7" onmouseover="rs('msualg_2/D7')" onmouseout="rh()">MSUALG_2:def 7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1_2"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="msualg_2.html#D7" onmouseover="rs('msualg_2/D7')" onmouseout="rh()">MSUALG_2:def 7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:19_2_1"/>
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E9:19_2_1"/>
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c8">U1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="msualg_2.html#D8" onmouseover="rs('msualg_2/D8')" onmouseout="rh()">MSUALG_2:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E2:19_2"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/><span class="kw">assume </span><a NAME="E4:19_2"/>
<font color="Maroon" title="c7">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="E5:19_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">GAA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c9">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" onmouseover="rs('msualg_2/D17')" onmouseout="rh()">MSUALG_2:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:19_2"/>
<font color="Maroon" title="c6">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:19"/>
<font color="Maroon" title="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" onmouseover="rs('msualg_2/D17')" onmouseout="rh()">MSUALG_2:def 17</a></span>;<br/>
<a NAME="E9:19"/><span class="kw">then </span>
 <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="c7">BB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="msualg_2.html#D17" onmouseover="rs('msualg_2/D17')" onmouseout="rh()">MSUALG_2:def 17</a></span>;<br/>
<a NAME="E10:19"/><span class="kw">then </span>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c7">BB</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T21" onmouseover="rs('msualg_2/T21')" onmouseout="rh()">MSUALG_2:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:19"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D4" onmouseover="rs('msafree/D4')" onmouseout="rh()">MSAFREE:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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