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<span class="kw">let </span><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">OU0</font> being  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU0</font> holds <br/> (  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Olive" title="b2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">OU0</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font> holds <br/> (  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:57"/>
 <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">o1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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">o1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> = <font color="Maroon" title="c4">o1</font> as   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font> ;<br/>
<a NAME="E2:57_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="c5">o</font>,<font color="Maroon" title="c2">OU0</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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE: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">S1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font></span>)</span> <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">S1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T33" target="_self" title="OSALG_2:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:57_1"/>
 <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</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">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E2:57"/>
( <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="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1">(<span class="default"><a href="osalg_2.html#K3" title="OSALG_2:func.3">OSConstants</a> <font color="Maroon" title="c2">OU0</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="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T29" target="_self" title="OSALG_2:th.29">Th29</a>, <a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:57"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="osalg_2.html#K9" title="OSALG_2:func.9">OSMSubSort</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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