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

<b>consider </b><font color="Maroon">L</font> being  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM1">Lattice</a>;<br/>
<a NAME="E2:10_1_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

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

<a NAME="E1:10_1_1_1"/>
( <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:10_1_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V16">complemented</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<a NAME="E4:10_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E5:10_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E6:10_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:10_1_1_2"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_2"/>
<font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a>
 <b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:10_1_1_3"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_3"/>
<font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a>
 <b>by </b><i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:10_1_1"/><b>then </b>
<font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<a NAME="E9:10_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V17">Boolean</a>
 
<b>by </b><i>, , <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>
<a NAME="E10:10_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">L</font> is <a href="robbins1.html#V6">join-idempotent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins1.html#D7">ROBBINS1:def 7</a><br/>

<b>thus </b><a NAME="E1:10_1_1_4"/>
<font color="Maroon">x</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:10_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:10_1_1_5"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_5"/>
<font color="Maroon">L</font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:10_1_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

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

<a NAME="E1:10_1_1_6"/>
( <font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:10_1_1_6"/>
<font color="Maroon">a</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:10_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V4" target="_self">complemented'</a>
 
<b>by </b><i/>;<br/>
<a NAME="E14:10_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<a NAME="E15:10_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V3" target="_self">distributive'</a>
 
<b>by </b><i/>;<br/>
<a NAME="E16:10_1_1"/>
<font color="Maroon">L</font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_7"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E2:10_1_1_7"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_7"/>
<font color="Maroon">L</font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:10_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V17">Boolean</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="robbins1.html#V6">join-idempotent</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="sheffer1.html#V4" target="_self">complemented'</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="sheffer1.html#V3" target="_self">distributive'</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V10">Lattice-like</a> )
 <b>by </b><i>, , <a class="ref" href="sheffer1.html#T1" target="_self">Th1</a>, , </i>;<br/>


</div>
