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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_2.html#V2" target="_self">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:52"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></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#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9" target="_self">Sub</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D15" target="_self">Def15</a></i>;<br/>
<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self">Def17</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></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#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E5:52_1"/><i><font color="Green">E39</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#R2">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E6:52_1"/><b>then </b><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E7:52_1"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E8:52_1"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E5:52_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E9:52_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E10:52_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> holds <br/>( the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="unialg_2.html#K5" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> ) ) )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E8:52_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E12:52_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E13:52_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></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#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E3:52_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="unialg_2.html#E4:52_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="unialg_2.html#E6:52_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="unialg_2.html#E7:52_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="unialg_2.html#E10:52_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="unialg_2.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:52"/>
 <a href="unialg_2.html#K11" target="_self">UniAlg_meet</a> <font color="Maroon">c<sub>1</sub></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>
