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

<span class="kw">let </span><font color="Maroon" title="c1">U0</font> be   <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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</a> <font color="Maroon" title="c1">U0</font>;<br/>
<a NAME="E1:45"/>
 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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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" 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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#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/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">B</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#K2" title="XBOOLE_0:func.2">\/</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#K2" title="XBOOLE_0:func.2">\/</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="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</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="c6">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c7">U2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U2</font> is   <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#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E5:45_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U3</font> is   <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#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">C</font> =  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#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U3</font> 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="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E7:45_1"/><span class="lab"><font color="Green" title="E31">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> is   <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#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">D</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#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U2</font> 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="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E9:45_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="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c9">B</font> 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="txt" href="#E7:45_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E11:45_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A6</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c7">U2</font></span>)</span> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U3</font> = 
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c11">D</font></span>)</span> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D13" target="_self" onmouseover="rs('unialg_2/D13')" onmouseout="rh()">Def13</a></span>
<br/>.= 
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c12">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="unialg_2.html#T20" target="_self" onmouseover="rs('unialg_2/T20')" onmouseout="rh()">Th20</a></span>
;<br/>
<a NAME="E13:45_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<a NAME="E14:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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> <font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</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#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">U2</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c8">U3</font></span>)</span> = 
<font color="Maroon" title="c6">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c10">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D13" target="_self" onmouseover="rs('unialg_2/D13')" onmouseout="rh()">Def13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c10">C</font></span>)</span> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c6">U1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T21" target="_self" onmouseover="rs('unialg_2/T21')" onmouseout="rh()">Th21</a></span>
<br/>.= 
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c12">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T20" target="_self" onmouseover="rs('unialg_2/T20')" onmouseout="rh()">Th20</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:45_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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#K10" title="UNIALG_2:func.10">UniAlg_join</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="#E3:45_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E14:45_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E12:45_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="unialg_2.html#D15" target="_self" onmouseover="rs('unialg_2/D15')" onmouseout="rh()">Def15</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="E2:45"/>
 <a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</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" onmouseover="rs('binop_1/D3')" onmouseout="rh()">BINOP_1:def 3</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>
