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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> implies ( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ) )</span><br/>

<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E34">A1</font></i>: 
<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">LL</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>)</span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>,<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>)</span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a>  ;<br/>
<a NAME="E3:54"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">LL</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">LL</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">LL</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">LL</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">LL</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">LL</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">LL</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">LL</font>) )
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:54"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="lattice2.html#T27" title="LATTICE2:th.27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29" title="LATTICE2:th.29">LATTICE2:29</a>, <a class="ref" href="lattice2.html#T31" title="LATTICE2:th.31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32" title="LATTICE2:th.32">LATTICE2:32</a>, <a class="ref" href="lattice2.html#T40" title="LATTICE2:th.40">LATTICE2:40</a>, <a class="ref" href="lattice2.html#T41" title="LATTICE2:th.41">LATTICE2:41</a></i>;<br/>
<a NAME="E4:54"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">L1</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L1</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">L2</font>) <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">L2</font>) )
 
<b>by </b><i><a class="ref" href="filter_1.html#T23" target="_self" title="FILTER_1:th.23">Th23</a>, <a class="ref" href="filter_1.html#T24" target="_self" title="FILTER_1:th.24">Th24</a>, <a class="ref" href="filter_1.html#T31" target="_self" title="FILTER_1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E5:54"/>
( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> )
 <b>by </b><i><a class="ref" href="lattice2.html#T17" title="LATTICE2:th.17">LATTICE2:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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