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

<span class="kw">set </span><font color="Maroon" title="c1">L</font> =  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ;<br/>
<span class="kw">thus </span><a NAME="E1:57_1_1"/>
 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a>,<a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:57_1_1"/>
 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:57_1_1"/>
 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  &amp;  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ) st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<span class="kw">take </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">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">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E1:57_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:57_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>; <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">thus </span><a NAME="E4:57_1_1"/>
 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2"><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>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</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="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">z</font></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="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</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="c4">z</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E1:57_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">z</font></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="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</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="c4">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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">thus </span><a NAME="E5:57_1_1"/>
 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  is  <a href="robbins1.html#V7" title="ROBBINS1:attr.7">well-complemented</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_3"><span class="kw">proof </span></a><div class="add">

<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 <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="robbins1.html#D10" target="_self" title="ROBBINS1:def.10">ROBBINS1:def 10</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="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E1:57_1_1_3"/>
( <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#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  &amp; <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#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:57_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>; <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>


</div>
