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

<b>let </b><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U0</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>;<br/>



<b>set </b><font color="Maroon" title="c3">o</font> =  <a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font>;<br/>
<b>set </b><font color="Maroon" title="c4">m</font> =  <a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font>;<br/>
<a NAME="E1:80"/><i><font color="Green" title="E55">A1</font></i>: 
 <a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T33" title="MSUALG_2:th.33">MSUALG_2:33</a></i>;<br/>
<a NAME="E2:80"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font> holds  <span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span>,<font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="osalg_2.html#K13" title="OSALG_2:func.13">OSSub</a> <font color="Maroon" title="c2">U0</font>;<br/>

<a NAME="E1:80_1"/><i><font color="Green" title="E56">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font> &amp; <span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="ref" href="osalg_2.html#T48" target="_self" title="OSALG_2:th.48">Th48</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span>
<b>by </b><i><a class="ref" href="osalg_2.html#T48" target="_self" title="OSALG_2:th.48">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K16" title="MSUALG_2:func.16">MSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c7">z</font>
<b>by </b><i><a class="txt" href="osalg_2.html#E1:80"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c7">z</font>
<b>by </b><i><a class="txt" href="osalg_2.html#E1:80_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="osalg_2.html#T48" target="_self" title="OSALG_2:th.48">Th48</a></i>
;<br/>
<b>hence </b><a NAME="E3:80_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c7">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
 <a href="osalg_2.html#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Maroon" title="c2">U0</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>;<br/>


</div>
