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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="sheffer1.html#V9" target="_self">properly_defined</a> <a href="sheffer1.html#V10" target="_self">satisfying_Sheffer_1</a> <a href="sheffer1.html#V11" target="_self">satisfying_Sheffer_2</a> <a href="sheffer1.html#V12" target="_self">satisfying_Sheffer_3</a>  <a href="sheffer1.html#L3" target="_self">ShefferOrthoLattStr</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>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>



<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">y</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">z</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</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="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span> <a href="sheffer1.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">z</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span> <a href="sheffer1.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">z</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.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">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="sheffer1.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">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="sheffer1.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">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">z</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E2:58"/>
<font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">z</font></span>)</span>
 ;<br/>


</div>
