<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">L</font> =  <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a> ;<br/>
<b>thus </b><a NAME="E1:48_3_1"/>
 <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  is <a href="sheffer1.html#V9" target="_self">properly_defined</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_3_1_1"/>
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  holds  <font color="Olive">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="robbins1.html#K3">`</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  holds  <font color="Olive">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">x</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">y</font></span>)</span> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  holds  <font color="Olive">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">y</font></span>)</span> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  holds  <font color="Olive">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> ) )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:48_3_1_1"/>
 <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  is <a href="sheffer1.html#V9" target="_self">properly_defined</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:48_3_1"/>
 <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  is <a href="robbins1.html#V7">well-complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins1.html#D10">ROBBINS1:def 10</a><br/>

<a NAME="E1:48_3_1_2"/>
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:48_3_1_2"/>
<font color="Maroon">a</font> <a href="robbins1.html#K3">`</a>  <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_3_1"/>
 <a href="sheffer1.html#K4" target="_self">TrivShefferOrthoLattStr</a>  is <a href="robbins1.html#V7">well-complemented</a>
 ;<br/>


</div>
