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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:54"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is   <a href="lattices.html#NM1">Lattice</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span><a href="filter_1.html#K6" target="_self">:|</a></span>,<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span><a href="filter_1.html#K6" target="_self">:|</a></span> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  ;<br/>
<a NAME="E3:54"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) )
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:54"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattice2.html#T27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29">LATTICE2:29</a>, <a class="ref" href="lattice2.html#T31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32">LATTICE2:32</a>, <a class="ref" href="lattice2.html#T40">LATTICE2:40</a>, <a class="ref" href="lattice2.html#T41">LATTICE2:41</a></i>;<br/>
<a NAME="E4:54"/><b>then </b>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="lattice2.html#R1">absorbs</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 
<b>by </b><i><a class="ref" href="filter_1.html#T23" target="_self">Th23</a>, <a class="ref" href="filter_1.html#T24" target="_self">Th24</a>, <a class="ref" href="filter_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>hence </b><a NAME="E5:54"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM1">Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM1">Lattice</a> )
 <b>by </b><i><a class="ref" href="lattice2.html#T17">LATTICE2:17</a></i>;<br/>


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