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<b>let </b><font color="Maroon">L</font> be  <a href="lattices.html#V11">distributive</a> <a href="lattices.html#V15">bounded</a> <a href="robbins1.html#V7" target="_self">well-complemented</a> <a href="robbins1.html#NM1">preOrthoLattice</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:61"/>
<font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:61"/><b>then </b><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<a NAME="E3:61"/>
<font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <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">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">y</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T7" target="_self">Th7</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T8" target="_self">Th8</a>, <a class="ref" href="lattices.html#T40">LATTICES:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T40">LATTICES:40</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>
;<br/>
<a NAME="E6:61"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T44">LATTICES:44</a></i>;<br/>
<a NAME="E7:61"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T44">LATTICES:44</a></i>;<br/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</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">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</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">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T31">LATTICES:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</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">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T8" target="_self">Th8</a>, , <a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T7" target="_self">Th7</a>, , <a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="lattices.html#T43">LATTICES:43</a></i>
;<br/>
<a NAME="E9:61"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:61"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<a NAME="E11:61"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="robbins1.html#D9" target="_self">Def9</a>, <a class="ref" href="robbins1.html#T9" target="_self">Th9</a>, <a class="ref" href="lattices.html#T32">LATTICES:32</a></i>;<br/>
<b>hence </b><a NAME="E12:61"/>
<font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
 <b>by </b><i><a class="txt" href="robbins1.html#E4:61"><i><font color="Green">E35</font></i></a></i>;<br/>


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