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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>   <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>   <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>   <a href="robbins1.html#V6" title="ROBBINS1:attr.6">join-idempotent</a>   <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a>   <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a>   <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a>   <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b1">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c3">y</font> iff <font color="Maroon" title="c2">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c3">y</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">y</font></span><br/><a NAME="E2:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#D6" target="_self" title="SHEFFER1:def.6">Def6</a></span>;<br/><a NAME="E3:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T20" target="_self" title="SHEFFER1:th.20">Th20</a></span>;<br/><a NAME="E4:33_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="sheffer1.html#D6" target="_self" title="SHEFFER1:def.6">Def6</a></span>;<br/><a NAME="E5:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T19" target="_self" title="SHEFFER1:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E6:33_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:33"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E3:33"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>;<br/>
<a NAME="E4:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T20" target="_self" title="SHEFFER1:th.20">Th20</a></span>;<br/>
<a NAME="E5:33"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>;<br/>
<a NAME="E6:33"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T19" target="_self" title="SHEFFER1:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:33"/>
<font color="Maroon" title="c2">x</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="sheffer1.html#D6" target="_self" title="SHEFFER1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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