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<div class="add">

<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="robbins1.html#L4" title="ROBBINS1:struct.4">OrthoLattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V10" title="ROBBINS1:attr.10">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/>

<span class="kw">thus </span><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#V10" title="ROBBINS1:attr.10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V10" title="ROBBINS1:attr.10">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/>

<span class="kw">assume </span><a NAME="E2:59"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V10" title="ROBBINS1:attr.10">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/>

<span class="kw">assume </span><a NAME="E3:59"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">L</font> is  <a href="robbins1.html#V10" title="ROBBINS1:attr.10">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/>

<span class="kw">assume </span><a NAME="E4:59"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="robbins1.html#V10" title="ROBBINS1:attr.10">de_Morgan</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E5:59"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="robbins3.html#D3" title="ROBBINS3:def.3">ROBBINS3:def 3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:59"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="robbins3.html#D3" title="ROBBINS3:def.3">ROBBINS3:def 3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E7:59"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E8:59"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="robbins4.html#D2" target="_self" title="ROBBINS4:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:59"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </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> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:59"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="robbins3.html#D1" title="ROBBINS3:def.1">ROBBINS3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:59"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </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"><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:59"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:59"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>
;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:59"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E45">A9</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E48">A12</font></span></a></span>
;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:59"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E49">A13</font></span></a></span>
;<br/>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E48">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:59"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
 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_11"><span class="kw">proof </span></a><div class="add">

<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="firebrick">::  thesis: </font></span></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_11"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><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"><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 class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E50">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:59"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
 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_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E49">A13</font></span></a></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> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E43">A7</font></span></a></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> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E49">A13</font></span></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="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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E50">A14</font></span></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="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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E50">A14</font></span></a></span>
;<br/>
<span class="kw">hence </span><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"><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 class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E50">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:59"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
 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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59"><span class="lab"><font color="Green" title="E52">A16</font></span></a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59"><span class="lab"><font color="Green" title="E51">A15</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:59"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E53">A17</font></span></a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E53">A17</font></span></a></span>
;<br/>
<span class="kw">hence </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:59"/>
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="robbins3.html#D2" title="ROBBINS3:def.2">ROBBINS3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E53">A17</font></span></a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="robbins3.html#D1" title="ROBBINS3:def.1">ROBBINS3:def 1</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E53">A17</font></span></a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L9</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#L4" title="ROBBINS1:struct.4">OrthoLattStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E19:59"><span class="lab"><font color="Green" title="E54">A18</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">L9</font> = <font color="Maroon" title="c2">L9</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#V10" title="ROBBINS1:attr.10">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#L4" title="ROBBINS1:struct.4">OrthoLattStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E42">A6</font></span></a></span>;<br/>
<a NAME="E23:59"/>
<font color="Maroon" title="c3">L9</font> is  <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E24:59"/>
<font color="Maroon" title="c1">L</font> is   <a href="robbins4.html#NM1" title="ROBBINS4:NM.1">Orthomodular_Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="robbins3.html#D6" title="ROBBINS3:def.6">ROBBINS3:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
