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<span class="kw">let </span><font color="Maroon" title="c1">U0</font> be   <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a>  <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">Universal_Algebra</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">  <a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">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="c2">o</font> =  <a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font>;<br/>
<a NAME="E1:46"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">U0</font> holds  <span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">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="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">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="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <font color="Maroon" title="c1">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="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">U1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">U2</font> = <font color="Maroon" title="c4">y</font> as   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">strict</a>   <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D14" target="_self" onmouseover="rs('unialg_2/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<a NAME="E2:46_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c6">U2</font> &amp; <span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c5">U1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D16" target="_self" onmouseover="rs('unialg_2/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<a NAME="E3:46_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T17" target="_self" onmouseover="rs('unialg_2/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E4:46_1"/><span class="kw">then </span>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c5">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U1</font> &amp; (  for <font color="Olive" title="b1">B2</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Olive" title="b1">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c5">U1</font></span>)</span> holds <br/>(  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c5">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> (<font color="Maroon" title="c1">U0</font>,<font color="Olive" title="b1">B2</font>) &amp; <font color="Olive" title="b1">B2</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>  ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D9" target="_self" onmouseover="rs('unialg_2/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="unialg_2.html#D9" target="_self" onmouseover="rs('unialg_2/D9')" onmouseout="rh()">Def9</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:46"/>
 <a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <font color="Maroon" title="c1">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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