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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V4</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c3">Y</font> = <font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<a NAME="E1:12"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> st <br/>(  <a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</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="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="equation.html#K1" title="EQUATION:func.1">""</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="msafree3.html#D2" target="_self" title="MSAFREE3:def.2">Def2</a></i>;<br/>
<a NAME="E2:12"/><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="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green" title="E8">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="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<i><font color="Green" title="E9">A2</font></i>: <font color="Maroon" title="c1">S</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
 <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msaterm.html#D1" title="MSATERM:def.1">MSATERM:def 1</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p4">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msafree.html#T12" title="MSAFREE:th.12">MSAFREE:12</a></i>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E5:12"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:12"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E12">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:12"/><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="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="msafree3.html#E3:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="msafree3.html#E7:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:12"/>
<font color="Maroon" title="c4">x</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msafree3.html#E4:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="msafree3.html#E5:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="msafree3.html#E7:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="msafree3.html#E8:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>


</div>
