<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</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="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:56"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E3:56"/><b>then </b><i><font color="Green">E41</font></i>: 
<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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<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>6</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>7</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> 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" target="_self">Def10</a></i>;<br/>
<a NAME="E5:56"/>
( <font color="Maroon">c<sub>7</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; <font color="Maroon">c<sub>6</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="E6:56"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>6</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> = <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#K2">\/</a> <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="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E8:56"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E10:56"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:56"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E3:56"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pboole.html#T22">PBOOLE:22</a></i>;<br/>
<a NAME="E12:56"/><b>then </b>
<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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E1:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msualg_2.html#E10:56"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E13:56"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E14:56"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font> is  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E15:56"/>
( <font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="msualg_2.html#E1:56"><i><font color="Green">E40</font></i></a>, <a class="ref" href="msualg_2.html#D18" target="_self">Def18</a>, <a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E16:56"/><b>then </b>
( <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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E17:56"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E18:56"/><b>then </b><i><font color="Green">E46</font></i>: 
<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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E3:56"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pboole.html#T19">PBOOLE:19</a></i>;<br/>
<a NAME="E19:56"/><i><font color="Green">E47</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K11" target="_self">/\</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E20:56"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K11" target="_self">/\</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E18:56"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E21:56"/><b>then </b>
 <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K11" target="_self">/\</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E22:56"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K11" target="_self">/\</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E23:56"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E19:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E24:56"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T17">PBOOLE:17</a></i>;<br/>
<a NAME="E25:56"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#K3">/\</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E23:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pboole.html#D13">PBOOLE:def 13</a></i>;<br/>
<a NAME="E26:56"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T17">PBOOLE:17</a></i>;<br/>
<a NAME="E27:56"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_2.html#E17:56"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<a NAME="E28:56"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E29:56"/><b>then </b>
 <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>8</sub></font> is  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of  <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E30:56"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="msualg_2.html#E8:56"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msualg_2.html#E14:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msualg_2.html#T8" target="_self">Th8</a></i>;<br/>


</div>
