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<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>, <font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E18</font></i>: 
 <a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L2</font>), <font color="Maroon">m</font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">L2</font>) as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L1</font>) <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">xy</font> = <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><a NAME="E3:13_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/><a NAME="E4:13_1"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xy</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L2</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">xy</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L2</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ab</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">ab</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E5:13_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">xy</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">xy</font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E6:13_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">xy</font> )
 <b>by </b><i><a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/><a NAME="E7:13_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">ab</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E8:13_1"/><b>then </b>
( <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L1</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">j</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><b>hence </b><a NAME="E9:13_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L1</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:13"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">xy</font> = <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><a NAME="E3:13_2"/>
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/><a NAME="E4:13_2"/><b>then </b>
( <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">x</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L2</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">y</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L2</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">ab</font>,<font color="Maroon">a</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font> &amp; <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">ab</font>,<font color="Maroon">b</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E5:13_2"/><b>then </b>
( <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">xy</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">xy</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E6:13_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">ab</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">xy</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/><a NAME="E7:13_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">ab</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E8:13_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ab</font> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">L1</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">m</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><b>hence </b><a NAME="E9:13_2"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">L1</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L1</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L1</font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L2</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L2</font> #)
 <b>by </b><i>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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