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<b>let </b><font color="Maroon">S</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">U0</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">o</font> =  <a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font>;<br/>
<a NAME="E1:65"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">U0</font> holds  <span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">y</font>,<font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">U0</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">U0</font>;<br/>

<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font> as  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:65_1"/><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">U2</font> &amp; <span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">U1</font> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>
<a NAME="E3:65_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> &amp; 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">U0</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:65_1"/><b>then </b>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E5:65_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18">PBOOLE:18</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">U1</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:65_1"/>
( <font color="Maroon">U1</font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">B</font> &amp; <font color="Maroon">U2</font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12" target="_self">GenMSAlg</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:65_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:65"/>
 <a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


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