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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:57"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:57"/><i><font color="Green">E35</font></i>: 
(  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="osalg_2.html#T34" target="_self">Th34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="msualg_2.html#K10">|</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:57"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_2.html#D16">MSUALG_2:def 16</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>3</sub></font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> is   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<a NAME="E1:57_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E2:57"><i><font color="Green">E35</font></i></a>, <a class="txt" href="osalg_2.html#E4:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E2:57_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E1:57"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_2.html#D2" target="_self">Def2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:57_1"/>
<font color="Maroon">c<sub>7</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="osalg_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:57_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></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/>
<b>assume </b><a NAME="E7:57_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E8:57_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E9:57_1"/>
 <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E10:57_1"/><b>then </b>
 <a href="osalg_2.html#K3" target="_self">OSConstants</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:57_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K5" target="_self">OSSubSort</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E6:57_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="osalg_2.html#E8:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="osalg_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E12:57_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:57_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:57_1_1"/><i><font color="Green">E41</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E4:57"><i><font color="Green">E36</font></i></a>, <a class="ref" href="osalg_2.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:57_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="osalg_2.html#K8" target="_self">OSSubSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_2.html#E11:57_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="osalg_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E5:57_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E3:57_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:57_1"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#T9">MSUALG_2:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:57"/>
 <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="msualg_2.html#K10">|</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="osalg_2.html#D13" target="_self">Def13</a></i>;<br/>

<b>thus </b><a NAME="E7:57"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="osalg_2.html#K9" target="_self">OSMSubSort</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="osalg_2.html#E4:57"><i><font color="Green">E36</font></i></a>, <a class="txt" href="osalg_2.html#E5:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="osalg_2.html#D13" target="_self">Def13</a></i>;<br/>


</div>
