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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="latwal_1.html#NM2" title="LATWAL_1:NM.2">WA_Lattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a>  implies <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E13">A0</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> </span><br/>

<a NAME="E2:30_1_1"/>
( <font color="Maroon" title="c1">L</font> is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">join-idempotent</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V5" title="SHEFFER1:attr.5">meet-idempotent</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V1" title="LATWAL_1:attr.1">satisfying_W3</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V2" title="LATWAL_1:attr.2">satisfying_W3'</a>  )
 
;<br/>
<a NAME="E3:30_1_1"/><span class="kw">then </span>
( (  for <font color="Olive" title="b1">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">v0</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v0</font> ) &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b2">v0</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">v0</font> ) &amp; (  for <font color="Olive" title="b1">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v0</font> ) &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b2">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v0</font> ) &amp; (  for <font color="Olive" title="b1">v2</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v2</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v1</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> ) &amp; (  for <font color="Olive" title="b1">v2</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">v0</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">v1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">v1</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> ) &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b2">v0</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v0</font> ) &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b2">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v0</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v0</font> ) &amp; (  for <font color="Olive" title="b1">v0</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">v1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v2</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E13">A0</font></span></a>, <a class="ref" href="robbins3.html#D3" onmouseover="rs('robbins3/D3')" onmouseout="rh()">ROBBINS3:def 3</a>, <a class="ref" href="lattices.html#D9" onmouseover="rs('lattices/D9')" onmouseout="rh()">LATTICES:def 9</a></span>;<br/>
<a NAME="E4:30_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">v0</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">v1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v0</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">v2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latwal_1.html#T18" target="_self" onmouseover="rs('latwal_1/T18')" onmouseout="rh()">WALDistri</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_1_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" onmouseover="rs('lattices/D5')" onmouseout="rh()">LATTICES:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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