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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="robbins1.html#V7" target="_self">well-complemented</a> <a href="robbins1.html#NM1">preOrthoLattice</a>;<br/>

<b>assume </b><a NAME="E1:110_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V17">Boolean</a>
 ;<br/>

<a NAME="E2:110_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V15">bounded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V16">complemented</a> )
 
<b>by </b><i><a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="lattices.html#V17">Boolean</a> <a href="robbins1.html#NM1">preOrthoLattice</a> <b>by </b><i><a class="txt" href="robbins1.html#E1:110_1_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E4:110_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T43">LATTICES:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="robbins1.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattices.html#T31">LATTICES:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="robbins1.html#T33" target="_self">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="robbins1.html#T34" target="_self">Th34</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="robbins1.html#T33" target="_self">Th33</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="robbins1.html#T34" target="_self">Th34</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins1.html#E2:110_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattices.html#T31">LATTICES:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T61" target="_self">Th61</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="robbins1.html#E4:110_1_1"><i><font color="Green">E68</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:110_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="robbins1.html#V5" target="_self">Huntington</a>
 <b>by </b><i><a class="ref" href="robbins1.html#D6" target="_self">Def6</a></i>;<br/>


</div>
