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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">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Olive" title="b1">U0</font> holds <br/> (  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Olive" title="b2">A</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; <font color="Olive" title="b2">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Olive" title="b2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">U0</font> be    <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"> for <font color="Olive" title="b1">A</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> holds <br/> (  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Olive" title="b1">A</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; <font color="Olive" title="b1">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Olive" title="b1">A</font> )</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> )</span><br/>





<b>thus </b><a NAME="E1:47"/>
 <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">MSUALG_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c4">o</font></span><br/>

<a NAME="E1:47_1"/>
 <a href="relat_1.html#K2" title="RELAT_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="c4">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> <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="c4">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> <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="c4">o</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T20" target="_self" title="MSUALG_2:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E2:47_1"/>
 <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c4">o</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6" target="_self" title="MSUALG_2:def.6">Def6</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>

<a NAME="E2:47"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></i>;<br/>
<a NAME="E3:47"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T16" target="_self" title="MSUALG_2:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_2.html#K9" title="MSUALG_2:func.9">MSSubSort</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E2:47"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</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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