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<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</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="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c3">o</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="E1:68"/>
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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> holds  <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#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="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#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="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#K4" title="BINOP_1:func.4">.</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#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="68_1"><b>proof </b></a><div class="add">

<b>let </b><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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font>;<br/>

<b>reconsider </b><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#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">u23</font> = <font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font>, <font color="Maroon" title="c11">u12</font> = <font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c8">U2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<i><font color="Green" title="E51">A1</font></i>: <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#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="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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">u23</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
<br/>.= 
<font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
;<br/>
<i><font color="Green" title="E52">A2</font></i>: <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#K4" title="BINOP_1:func.4">.</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#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="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#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c11">u12</font>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c8">U2</font></span>)</span> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
;<br/>
<a NAME="E5:68_1"/><i><font color="Green" title="E53">A3</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c8">U2</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></i>;<br/>
<a NAME="E6:68_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">U3</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></i>;<br/>
<a NAME="E7:68_1"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Olive" title="b1">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font></span>)</span></span>)</span> holds <br/>( <font color="Olive" title="b1">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">U1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">U2</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">U3</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b1">B</font> ) ) )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">C</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U2</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">U3</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></i>;<br/>
<a NAME="E9:68_1"/>
<font color="Maroon" title="c12">C</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c7">U1</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">U2</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">U3</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T35" title="PBOOLE:th.35">PBOOLE:35</a></i>;<br/>
<b>hence </b><a NAME="E10:68_1"/>
<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#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="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#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="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#K4" title="BINOP_1:func.4">.</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#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>
 <b>by </b><i><a class="txt" href="msualg_2.html#E3:68_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="msualg_2.html#E4:68_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="msualg_2.html#E5:68_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="msualg_2.html#E7:68_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="msualg_2.html#D17" target="_self" title="MSUALG_2:def.17">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68"/>
 <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="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>;<br/>


</div>
