<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="robbins3.html#NM1" title="ROBBINS3:NM.1">Ortholattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="robbins4.html#V1" title="ROBBINS4:attr.1">orthomodular</a>  iff for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:56"/>
( <font color="Maroon" title="c1">L</font> is  <a href="robbins4.html#V1" title="ROBBINS4:attr.1">orthomodular</a>  implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> ) implies <font color="Maroon" title="c1">L</font> is  <a href="robbins4.html#V1" title="ROBBINS4:attr.1">orthomodular</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="robbins4.html#V1" title="ROBBINS4:attr.1">orthomodular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span></span><br/>

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span></span><br/>

<a NAME="E3:56_1"/>
( <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T3" title="FILTER_0:th.3">FILTER_0:3</a>, <a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></span>;<br/>
<a NAME="E4:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></span>;<br/>
<a NAME="E5:56_1"/>
( <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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> &amp; <font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T3" title="FILTER_0:th.3">FILTER_0:3</a>, <a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></span>;<br/>
<a NAME="E6:56_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></span>;<br/>
<a NAME="E7:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></span>;<br/>
<a NAME="E8:56_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></span>;<br/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="robbins4.html#T36" target="_self" title="ROBBINS4:th.36">Th36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:56_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E8:56_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:56"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="robbins4.html#V1" title="ROBBINS4:attr.1">orthomodular</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</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="Red">:: according to </font></span><a class="ref" href="robbins4.html#D1" target="_self" title="ROBBINS4:def.1">ROBBINS4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">y</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E3:56"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E37">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></span>
;<br/>
<span class="kw">hence </span><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> = 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
