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<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="osalg_1.html#V13">monotone</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">U2</font> be   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U1</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a><br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">o1</font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D23">OSALG_1:def 23</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green">E34</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U1</font> &amp; ( for <font color="Olive">B</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U1</font>  st <font color="Olive">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> holds <br/>( <font color="Olive">B</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U2</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U1</font>,<font color="Olive">B</font> ) ) )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:19"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> as    <a href="osalg_2.html#M2">OSSubset</a> of <font color="Maroon">U1</font> <b>by </b><i>, <a class="ref" href="osalg_2.html#D2">OSALG_2:def 2</a></i>;<br/>
<a NAME="E6:19"/>
<font color="Maroon">B</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">B</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">B</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o2</font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<i><font color="Green">E40</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">B</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>
<br/>.= 
<font color="Maroon">o1</font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
;<br/>
<i><font color="Green">E44</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">U2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">B</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>
<br/>.= 
<font color="Maroon">o2</font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
;<br/>
<a NAME="E10:19"/><i><font color="Green">E45</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#T26">OSALG_1:26</a></i>;<br/>
<a NAME="E11:19"/><i><font color="Green">E50</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="msafree3.html#T38">MSAFREE3:38</a></i>;<br/>
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
;<br/>
<b>hence </b><a NAME="E13:19"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">U2</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">U2</font>
 ;<br/>


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