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<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K15" title="OSALG_2:func.15">OSAlg_meet</a> <font color="Olive" title="b1">U0</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#V1" title="BINOP_1:attr.1">commutative</a> </span><br/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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="E1:78"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <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#V1" title="BINOP_1:attr.1">commutative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T31" onmouseover="rs('msualg_2/T31')" onmouseout="rh()">MSUALG_2:31</a></span>;<br/>
<a NAME="E2:78"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <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#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K4" title="BINOP_1:func.4">.</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="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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span><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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</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="c5">x</font>,<font color="Maroon" title="c6">y</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T43" target="_self" onmouseover="rs('osalg_2/T43')" onmouseout="rh()">Th43</a></span>
<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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="binop_1.html#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</a></span>
<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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#T43" target="_self" onmouseover="rs('osalg_2/T43')" onmouseout="rh()">Th43</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:78_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#K4" title="BINOP_1:func.4">.</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="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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:78"/>
 <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#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D2" onmouseover="rs('binop_1/D2')" onmouseout="rh()">BINOP_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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