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<b>set </b><font color="Maroon" title="c1">L</font> =  <a href="robbins1.html#K2" title="ROBBINS1:func.2">TrivOrtLat</a> ;<br/>
<b>thus </b><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>
  <i><font color="Red">:: according to </font></i><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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c2">c</font> being   <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> ;<br/>
<b>take </b>
<font color="Maroon" title="c2">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a><br/>

<b>thus </b><a NAME="E2:57_1_1_1"/>
for <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> ) holds <br/> ( <font color="Maroon" title="c2">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><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>
 ;<br/>

<b>thus </b><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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a><br/>

<b>take </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">b</font>;<br/>

<a NAME="E1:57_1_1_2"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:57_1_1_2_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="c2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">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; <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>  &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">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>  )
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_1_1_2"/>
<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>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a><br/>

<b>thus </b><a NAME="E1:57_1_1_3"/>
<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>
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins1.html#D10" target="_self" title="ROBBINS1:def.10">ROBBINS1:def 10</a><br/>

<b>thus </b><a NAME="E1:57_1_1_4"/>
<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:57_1_1_4_1"/>
( <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; <font color="Maroon" title="c2">a</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> <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>  &amp; <font color="Maroon" title="c2">a</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="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>  )
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div>
