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

<b>let </b><font color="Maroon">A</font> be  <a href="lattice3.html#V1" target="_self">with_suprema</a> <a href="lattice3.html#V2" target="_self">with_infima</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Olive">y'</font>;<br/>
<a NAME="E1:46"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y'</font>
;<br/>

<b>thus </b><a NAME="E2:46_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y'</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">j</font> being   <a href="binop_1.html#NM2">BinOp</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:46"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="lattice3.html#E4:46"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Olive">y'</font>;<br/>
<a NAME="E4:46"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y'</font>
;<br/>

<b>thus </b><a NAME="E2:46_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y'</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</font> being   <a href="binop_1.html#NM2">BinOp</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:46"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #);<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E2:46_3"/>
( <font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">j</font> <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">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:46_3"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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">z</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<font color="Maroon">j</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice3.html#E3:46"><i><font color="Green">E22</font></i></a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><a NAME="E2:46_6"/>
( <font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">m</font> <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">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:46_6"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_7"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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">z</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_8"><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 <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> = 
<font color="Maroon">m</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E13:46"/><b>then </b>
(  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V4">join-commutative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V5">join-associative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V8">meet-absorbing</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V6">meet-commutative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V7">meet-associative</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i>, , <a class="txt" href="lattice3.html#E7:46"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,<font color="Maroon">j</font>,<font color="Maroon">m</font> #) as  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<a NAME="E15:46"/><i><font color="Green">E97</font></i>: 
 <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="filter_1.html#K9">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="filter_1.html#K9">]</a></span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8">FILTER_1:def 8</a></i>;<br/>
<a NAME="E16:46"/>
 <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E1:46_9"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_9_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:46_9_1"/><i><font color="Green">E98</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_1.html#K9">]</a></span> &amp; <font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<font color="Maroon">p'</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">q'</font> = 
<font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/>
<a NAME="E6:46_9_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:46_9_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:46_9"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:46_9"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E7:46_9"/>
( <font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:46_9"/><b>then </b>
( <font color="Maroon">x1</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">x2</font> )
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">x2</font> = 
<font color="Maroon">x1</font> <a href="lattice3.html#K13" target="_self">"\/"</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>
<br/>.= 
<font color="Maroon">y1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="lattice3.html#E6:46"><i><font color="Green">E26</font></i></a></i>
;<br/>
<a NAME="E10:46_9"/><b>then </b>
<font color="Maroon">y1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:46_9"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:46"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L</font>
 ;<br/>


</div>
