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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<b>thus </b><a NAME="E1:35"/>
( <font color="Maroon">c<sub>1</sub></font> is  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V3" target="_self">distributive'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V4" target="_self">complemented'</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a> ;<br/>
<a NAME="E3:35_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E1:35_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D3" target="_self">Def3</a></i>;<br/>

<a NAME="E5:35_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E1:35_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="lattices.html#T43">LATTICES:43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D1" target="_self">Def1</a></i>;<br/>

<b>thus </b><a NAME="E7:35_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> )
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:35_1"><i><font color="Green">E34</font></i></a></i>;<br/>

<a NAME="E8:35_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E9:35_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#T19">LATTICES:19</a></i>;<br/>
<b>hence </b><a NAME="E10:35_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V3" target="_self">distributive'</a> )
 <b>by </b><i><a class="ref" href="sheffer1.html#D5" target="_self">Def5</a></i>;<br/>

<b>thus </b><a NAME="E11:35_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V4" target="_self">complemented'</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:35_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="sheffer1.html#E4:35_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sheffer1.html#E6:35_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sheffer1.html#E10:35_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sheffer1.html#T25" target="_self">Th25</a></i>;<br/>


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

<b>thus </b><a NAME="E2:35"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V3" target="_self">distributive'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V4" target="_self">complemented'</a> implies <font color="Maroon">c<sub>1</sub></font> is  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V3" target="_self">distributive'</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="sheffer1.html#V4" target="_self">complemented'</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V4">join-commutative</a> <a href="lattices.html#V6">meet-commutative</a> <a href="lattices.html#V11">distributive</a> <a href="robbins1.html#V6">join-idempotent</a> <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> <a href="sheffer1.html#V3" target="_self">distributive'</a> <a href="sheffer1.html#V4" target="_self">complemented'</a>  <a href="lattices.html#L3">LattStr</a>  <b>by </b><i><a class="ref" href="sheffer1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:35_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E4:35_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:35_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V15">bounded</a>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E3:35_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<a NAME="E6:35_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V16">complemented</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:35_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="ref" href="sheffer1.html#T11" target="_self">Th11</a>, <a class="ref" href="sheffer1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E8:35_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V5">join-associative</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E9:35_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V7">meet-associative</a>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E7:35_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sheffer1.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E10:35_2"/>
<font color="Maroon">c<sub>1</sub></font> is  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E5:35_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sheffer1.html#E6:35_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sheffer1.html#E7:35_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sheffer1.html#E8:35_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lattices.html#D10">LATTICES:def 10</a>, <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>


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


</div>
