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<span class="kw">let </span><font color="Maroon" title="c1">L</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" 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="latwal_1.html#V4" title="LATWAL_1:attr.4">satisfying_WAL31</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V5" title="LATWAL_1:attr.5">satisfying_WAL32</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V6" title="LATWAL_1:attr.6">satisfying_WAL33</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V7" title="LATWAL_1:attr.7">satisfying_WAL34</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V8" title="LATWAL_1:attr.8">satisfying_WAL35</a>  implies ( <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>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E31">a1</font></span>: 
( <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V4" title="LATWAL_1:attr.4">satisfying_WAL31</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V5" title="LATWAL_1:attr.5">satisfying_WAL32</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V6" title="LATWAL_1:attr.6">satisfying_WAL33</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V7" title="LATWAL_1:attr.7">satisfying_WAL34</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="latwal_1.html#V8" title="LATWAL_1:attr.8">satisfying_WAL35</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="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>  )</span><br/>

<a NAME="E2:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 for <font color="Olive" title="b1">v0</font>, <font color="Olive" title="b2">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#K2" title="LATTICES:func.2">"/\"</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> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">v0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latwal_1.html#T21" target="_self" onmouseover="rs('latwal_1/T21')" onmouseout="rh()">Lemma3</a></span>;<br/>
<a NAME="E3:55_1_1"/>
 for <font color="Olive" title="b1">v0</font>, <font color="Olive" title="b2">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#K1" title="LATTICES:func.1">"\/"</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> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">v0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E31">a1</font></span></a>, <a class="ref" href="latwal_1.html#T22" target="_self" onmouseover="rs('latwal_1/T22')" onmouseout="rh()">Lemma4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:55_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E31">a1</font></span></a>, <a class="ref" href="latwal_1.html#T20" target="_self" onmouseover="rs('latwal_1/T20')" onmouseout="rh()">Lemma2</a>, <a class="ref" href="lattices.html#D4" onmouseover="rs('lattices/D4')" onmouseout="rh()">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D6" onmouseover="rs('lattices/D6')" onmouseout="rh()">LATTICES:def 6</a>, <a class="txt" href="latwal_1.html#E42"><span class="lab"><font color="Green" title="E27">Lemma1</font></span></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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