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

<b>let </b><font color="Maroon">U0</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon">o</font> =  <a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font>;<br/>
<a NAME="E1:50"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">U0</font> holds  <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">y</font>,<font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span>,<font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">U0</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">U0</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">U0</font>;<br/>

<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">x</font>, <font color="Maroon">U2</font> = <font color="Maroon">y</font>, <font color="Maroon">U3</font> = <font color="Maroon">z</font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="txt" href="unialg_2.html#E10:50_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E2:50_1"/>
( <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font> &amp; <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:50_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U2</font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:50_1"/><i><font color="Green">E26</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:50_1"/><b>then </b><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<i><font color="Green">E30</font></i>: <font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">U2</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font></span>)</span> = 
<font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">C</font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U1</font>
<b>by </b><i/>
<br/>.= 
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">B</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E5:50_1"><i><font color="Green">E26</font></i></a></i>
;<br/>
<a NAME="E11:50_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">U1</font> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U2</font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font> = 
<span class="p1">(<span class="default"><a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">D</font></span>)</span> <a href="unialg_2.html#K8" target="_self">"\/"</a> <font color="Maroon">U3</font>
<b>by </b><i/>
<br/>.= 
 <a href="unialg_2.html#K7" target="_self">GenUnivAlg</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="txt" href="unialg_2.html#E5:50_1"><i><font color="Green">E26</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:50_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50"/>
 <a href="unialg_2.html#K10" target="_self">UniAlg_join</a> <font color="Maroon">U0</font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div>
