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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="sheffer1.html#V9" title="SHEFFER1:attr.9">properly_defined</a> <a href="sheffer1.html#V10" title="SHEFFER1:attr.10">satisfying_Sheffer_1</a> <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a> <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a>  <a href="sheffer1.html#L3" title="SHEFFER1:struct.3">ShefferOrthoLattStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a><br/>



<b>set </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D15" target="_self" title="SHEFFER1:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E2:58"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 ;<br/>


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