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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">I</font> is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span><br/>







<a NAME="E1:49"/>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/>
<a NAME="E2:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" target="_self" title="FILTER_0:def.8">Def8</a></span>;<br/>
<a NAME="E3:49"/>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/>
<a NAME="E4:49"/><span class="kw">then </span>
<font color="Maroon" title="c3">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" target="_self" title="FILTER_0:def.8">Def8</a></span>;<br/>
<a NAME="E5:49"/><span class="kw">then </span>
<font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="filter_0.html#T6" target="_self" title="FILTER_0:th.6">Th6</a></span>;<br/>
<a NAME="E6:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></span>;<br/>
<a NAME="E7:49"/>
( <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> &amp; <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a>, <a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></span>;<br/>
<a NAME="E8:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T6" target="_self" title="FILTER_0:th.6">Th6</a></span>;<br/>
<a NAME="E9:49"/>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" target="_self" title="FILTER_0:def.8">Def8</a></span>;<br/>
<a NAME="E10:49"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:49"/>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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