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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <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/>

<b>assume </b><a NAME="E1:64_1_2"/><i><font color="Green">E40</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:64_1_2"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> is   <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="E3:64_1_2"/><b>then </b>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:64_1_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</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="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>10</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>4</sub></font> as   <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/>
<a NAME="E7:64_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<a NAME="E8:64_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</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#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</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="E10:64_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_2.html#E1:64_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:64_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


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