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

<b>let </b><font color="Maroon">S1</font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U0</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S1</font>;<br/>



<b>set </b><font color="Maroon">o</font> =  <a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font>;<br/>
<a NAME="E1:78"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font> holds  <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">y</font>,<font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_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></span>)</span>,<font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</a> <font color="Maroon">U0</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="osalg_2.html#K13" target="_self">OSSub</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>, <font color="Maroon">U3</font> = <font color="Maroon">z</font> as  <a href="msualg_1.html#V4">strict</a> <a href="osalg_2.html#NM3">OSSubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="txt" href="osalg_2.html#E8:78_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E2:78_1"/>
( <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font> &amp; <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_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="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font> )
 
<b>by </b><i><a class="txt" href="osalg_2.html#E10:78_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:78_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_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></span>)</span>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font> )
 
<b>by </b><i><a class="txt" href="osalg_2.html#E10:78_1"><i><font color="Green">E48</font></i></a></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> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font></span>)</span>;<br/>
<a NAME="E4:78_1"/><i><font color="Green">E26</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> &amp; 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> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</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/>
<a NAME="E5:78_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( 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> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:78_1"/><b>then </b><i><font color="Green">E45</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font></span>)</span> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:78_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> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E8:78_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( 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> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</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="E9:78_1"/><b>then </b><i><font color="Green">E47</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</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/>
<a NAME="E10:78_1"/><i><font color="Green">E48</font></i>: 
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>reconsider </b><font color="Maroon">C1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</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/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C1</font> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#T11" target="_self">Th11</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">D1</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/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D1</font> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E15:78_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font></span>)</span> <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/>
<a NAME="E16:78_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> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font></span>)</span> is   <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/>
<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> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font></span>)</span> as    <a href="osalg_2.html#M2" target="_self">OSSubset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="osalg_2.html#D4" target="_self">Def4</a>, </i>;<br/>
<i><font color="Green">E73</font></i>: <font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font></span>)</span> = 
<font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">C</font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U1</font>
<b>by </b><i><a class="ref" href="osalg_2.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="osalg_2.html#T4" target="_self">Th4</a></i>
;<br/>
<a NAME="E19:78_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">B</font> <a href="pboole.html#R6">=</a> <font color="Maroon">D</font> <a href="pboole.html#K2">\/</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T34">PBOOLE:34</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U2</font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">D</font></span>)</span> <a href="osalg_2.html#K11" target="_self">"\/"_os</a> <font color="Maroon">U3</font>
<b>by </b><i/>
<br/>.= 
 <a href="osalg_2.html#K10" target="_self">GenOSAlg</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="osalg_2.html#T4" target="_self">Th4</a></i>
;<br/>
<b>hence </b><a NAME="E21:78_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" target="_self">OSAlg_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></span>)</span>,<font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:78"/>
 <a href="osalg_2.html#K14" target="_self">OSAlg_join</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div>
