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

<span class="kw">let </span><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U0</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font> holds   <a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Olive" title="b1">U0</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">o</font> =  <a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font>;<br/>
<a NAME="E1:82"/>
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#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">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#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">U1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">U2</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">U3</font> = <font color="Maroon" title="c6">z</font> as   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="osalg_2.html#NM3" title="OSALG_2:NM.3">OSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D15" target="_self" title="OSALG_2:def.15">Def15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">B</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span>;<br/>
<a NAME="E2:82_1"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<a NAME="E3:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></span>;<br/>
<a NAME="E4:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A2</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E5:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></span>;<br/>
<a NAME="E6:82_1"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E7:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A3</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">C1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E9:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<a NAME="E10:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A4</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="osalg_2.html#T2" target="_self" title="OSALG_2:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">C</font> = <font color="Maroon" title="c11">C1</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>;<br/>
<a NAME="E12:82_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<a NAME="E13:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A6</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="osalg_2.html#T2" target="_self" title="OSALG_2:th.2">Th2</a></span>;<br/>
<a NAME="E14:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></span>;<br/>
<a NAME="E15:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A7</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E16:82_1"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">D1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<a NAME="E18:82_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">U2</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c9">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D16" target="_self" title="OSALG_2:def.16">Def16</a></span>;<br/>
<a NAME="E19:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U1</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c9">U3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D16" target="_self" title="OSALG_2:def.16">Def16</a></span>;<br/>
<a NAME="E20:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:82_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="osalg_2.html#T2" target="_self" title="OSALG_2:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">D</font> = <font color="Maroon" title="c13">D1</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>;<br/>
<a NAME="E22:82_1"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U1</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c8">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D16" target="_self" title="OSALG_2:def.16">Def16</a></span>;<br/>
<a NAME="E23:82_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E24:82_1"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">B</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:82_1"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="osalg_2.html#D2" target="_self" title="OSALG_2:def.2">Def2</a></span>;<br/>
<a NAME="E26:82_1"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">D</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T34" title="PBOOLE:th.34">PBOOLE:34</a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A11</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c8">U2</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c9">U3</font> = 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c14">D</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c9">U3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D14" target="_self" title="OSALG_2:def.14">Def14</a></span>
<br/>.= 
 <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c15">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:82_1"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="osalg_2.html#T42" target="_self" title="OSALG_2:th.42">Th42</a></span>
;<br/>
<font color="Maroon" title="c7">U1</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c9">U3</font></span>)</span> = 
<font color="Maroon" title="c7">U1</font> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c12">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D14" target="_self" title="OSALG_2:def.14">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="osalg_2.html#K11" title="OSALG_2:func.11">"\/"_os</a> <font color="Maroon" title="c7">U1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T44" target="_self" title="OSALG_2:th.44">Th44</a></span>
<br/>.= 
 <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c15">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T42" target="_self" title="OSALG_2:th.42">Th42</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E29:82_1"/>
<span class="p1">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">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#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:82_1"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E19:82_1"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="txt" href="#E27:82_1"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="osalg_2.html#D16" target="_self" title="OSALG_2:def.16">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:82"/>
 <a href="osalg_2.html#K14" title="OSALG_2:func.14">OSAlg_join</a> <font color="Maroon" title="c2">U0</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
