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

<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#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#V2" title="BINOP_1:attr.2">associative</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="E1:53"/>
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="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>,<span class="p1">(<span class="default"><span class="p2">(<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="b3">z</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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>, <font color="Maroon" title="c5">z</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" 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="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>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">z</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Maroon" title="c5">z</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">U1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">U2</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">U3</font> = <font color="Maroon" title="c5">z</font> as   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">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#D15" target="_self" title="UNIALG_2:def.15">Def15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">u23</font> = <font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font>, <font color="Maroon" title="c10">u12</font> = <font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c7">U2</font> as    <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" target="_self" title="UNIALG_2:def.15">Def15</a></span>;<br/>
<span class="lab"><font color="Green" title="E36">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>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">z</font></span>)</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="c9">u23</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></span>
<br/>.= 
<font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></span>
;<br/>
<span class="lab"><font color="Green" title="E37">A2</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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Maroon" title="c5">z</font> = 
<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="c10">u12</font>,<font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c7">U2</font></span>)</span> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D17" target="_self" title="UNIALG_2:def.17">Def17</a></span>
;<br/>
<a NAME="E5:53_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U2</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="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T20" target="_self" title="UNIALG_2:th.20">Th20</a></span>;<br/>
<a NAME="E6:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A3</font></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="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</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="c7">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="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D10" target="_self" title="UNIALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E7:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A4</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">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="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T20" target="_self" title="UNIALG_2:th.20">Th20</a></span>;<br/>
<a NAME="E8:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
for <font color="Olive" title="b1">B</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">B</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">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font></span>)</span></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">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font></span>)</span></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">B</font> &amp; <font color="Olive" title="b1">B</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="txt" href="#E6:53_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="unialg_2.html#D10" target="_self" title="UNIALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E9:53_1"/><span class="lab"><font color="Green" title="E41">A6</font></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">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c8">U3</font></span>)</span></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">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">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="c8">U3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E7:53_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="unialg_2.html#D10" target="_self" title="UNIALG_2:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">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="c8">U3</font></span>)</span> as  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> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<a NAME="E11:53_1"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U1</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="c7">U2</font></span>)</span> <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="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E12:53_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">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="c7">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T20" target="_self" title="UNIALG_2:th.20">Th20</a></span>;<br/>
<a NAME="E13:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A8</font></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">U1</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c7">U2</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">U1</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="c7">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D10" target="_self" title="UNIALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E14:53_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">U1</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="c7">U2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T20" target="_self" title="UNIALG_2:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:53_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>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">z</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E13:53_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E9:53_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E8:53_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E11:53_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="unialg_2.html#D10" target="_self" title="UNIALG_2:def.10">Def10</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="E2:53"/>
 <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#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>
