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<div class="add">

<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="robbins1.html#L3" title="ROBBINS1:struct.3">OrthoLattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> iff ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> ) )</span><br/>

<b>thus </b><a NAME="E1:59"/>
( <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> implies <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> )</span><br/>

<b>assume </b><a NAME="E2:59"/><i><font color="Green" title="E37">z1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> or  not <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> or  not <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> or <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> )</span><br/>

<b>assume </b><a NAME="E3:59"/><i><font color="Green" title="E38">z2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> or  not <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> or <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> )</span><br/>

<b>assume </b><a NAME="E4:59"/><i><font color="Green" title="E39">z3</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a> or <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a> )</span><br/>

<b>assume </b><a NAME="E5:59"/><i><font color="Green" title="E40">z4</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a></span><br/>

<a NAME="E6:59"/><i><font color="Green" title="E41">a</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <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="c3">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:59"><i><font color="Green" title="E38">z2</font></i></a>, <a class="ref" href="robbins3.html#D3" title="ROBBINS3:def.3">ROBBINS3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:59"/><i><font color="Green" title="E42">b</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E6:59"><i><font color="Green" title="E41">a</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E3:59"><i><font color="Green" title="E38">z2</font></i></a>, <a class="ref" href="robbins3.html#D3" title="ROBBINS3:def.3">ROBBINS3:def 3</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:59"/><i><font color="Green" title="E43">c</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:59"><i><font color="Green" title="E40">z4</font></i></a>, <a class="ref" href="robbins4.html#D2" target="_self" title="ROBBINS4:def.2">DefMod3</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:59"/><i><font color="Green" title="E44">d</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E8:59"><i><font color="Green" title="E43">c</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="#E6:59"><i><font color="Green" title="E41">a</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:59"/><i><font color="Green" title="E45">e</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E1:59_5"/>
<font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <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#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:59"><i><font color="Green" title="E42">b</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:59_5"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E37">z1</font></i></a>, <a class="ref" href="robbins3.html#D1" title="ROBBINS3:def.1">ROBBINS3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:59"/><i><font color="Green" title="E46">f</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E10:59"><i><font color="Green" title="E45">e</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="#E9:59"><i><font color="Green" title="E44">d</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:59"/><i><font color="Green" title="E47">g</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E11:59"><i><font color="Green" title="E46">f</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E7:59"><i><font color="Green" title="E42">b</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:59"/><i><font color="Green" title="E48">h</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:59"><i><font color="Green" title="E45">e</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E12:59"><i><font color="Green" title="E47">g</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:59_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E7:59"><i><font color="Green" title="E42">b</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:59"/><i><font color="Green" title="E49">i</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:59"><i><font color="Green" title="E45">e</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E13:59"><i><font color="Green" title="E48">h</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:59_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E7:59"><i><font color="Green" title="E42">b</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:59"/><i><font color="Green" title="E50">j</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E9:59"><i><font color="Green" title="E44">d</font></i></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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E14:59"><i><font color="Green" title="E49">i</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:59_10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E13:59"><i><font color="Green" title="E48">h</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:59"/><i><font color="Green" title="E51">k</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="#E14:59"><i><font color="Green" title="E49">i</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E8:59"><i><font color="Green" title="E43">c</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E14:59"><i><font color="Green" title="E49">i</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E15:59"><i><font color="Green" title="E50">j</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E15:59"><i><font color="Green" title="E50">j</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:59_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E15:59"><i><font color="Green" title="E50">j</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E17:59"/><i><font color="Green" title="E52">l</font></i>: 
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  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_12"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span><br/>

<a NAME="E1:59_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:59"><i><font color="Green" title="E46">f</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:59_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </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="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E15:59"><i><font color="Green" title="E50">j</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E18:59"/><i><font color="Green" title="E53">m</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>thus </b><font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E16:59"><i><font color="Green" title="E51">k</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E17:59"><i><font color="Green" title="E52">l</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:59"/><i><font color="Green" title="E54">l3</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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#D9" title="LATTICES:def.9">LATTICES:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E18:59"><i><font color="Green" title="E53">m</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E9:59"><i><font color="Green" title="E44">d</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="#E18:59"><i><font color="Green" title="E53">m</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:59_14"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:59"/><i><font color="Green" title="E55">l1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V2" title="ROBBINS3:attr.2">meet-Associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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="robbins3.html#D2" title="ROBBINS3:def.2">ROBBINS3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E18:59"><i><font color="Green" title="E53">m</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E37">z1</font></i></a>, <a class="ref" href="robbins3.html#D1" title="ROBBINS3:def.1">ROBBINS3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E18:59"><i><font color="Green" title="E53">m</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  <a href="robbins1.html#L3" title="ROBBINS1:struct.3">OrthoLattStr</a>  <b>by </b><i><a class="txt" href="#E3:59"><i><font color="Green" title="E38">z2</font></i></a>, <a class="txt" href="#E20:59"><i><font color="Green" title="E55">l1</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E37">z1</font></i></a>, <a class="txt" href="#E19:59"><i><font color="Green" title="E54">l3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">L'</font> = <font color="Maroon" title="c2">L'</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a> <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> <a href="robbins4.html#V2" title="ROBBINS4:attr.2">Orthomodular</a>  <a href="robbins1.html#L3" title="ROBBINS1:struct.3">OrthoLattStr</a>  <b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">z3</font></i></a>, <a class="txt" href="#E5:59"><i><font color="Green" title="E40">z4</font></i></a></i>;<br/>
<a NAME="E23:59"/>
<font color="Maroon" title="c3">L'</font> is <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>
 
;<br/>
<b>hence </b><a NAME="E24:59"/>
<font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a>
 <b>by </b><i><a class="txt" href="#E18:59"><i><font color="Green" title="E53">m</font></i></a>, <a class="ref" href="robbins3.html#D6" title="ROBBINS3:def.6">ROBBINS3:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
