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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> iff <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )</span><br/>

<a NAME="E1:76"/><i><font color="Green" title="E28">A1</font></i>: 
( ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> ) ) &amp; ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> ) &amp; ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> implies ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> ) ) &amp; ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> ) &amp; ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> implies <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> ) &amp; ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> ) &amp; ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> ) &amp; ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> implies <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> ) )
 
<b>by </b><i><a class="ref" href="lattice2.html#T63" title="LATTICE2:th.63">LATTICE2:63</a>, <a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a></i>;<br/>
<b>thus </b><a NAME="E2:76"/>
( <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> implies <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> implies <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> st <font color="Olive" title="b1">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p'</font></span><br/><b>consider </b><font color="Maroon" title="c3">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:76_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a>  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c2">p'</font>
 <b>by </b><i><a class="txt" href="#E1:76_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/><b>take </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c3">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p'</font></span><br/><a NAME="E3:76_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c2">p'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c2">p'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="#E1:76_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E2:76_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/><a NAME="E4:76_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="lattice2.html#T78" title="LATTICE2:th.78">LATTICE2:78</a>, <a class="ref" href="lattice2.html#T79" title="LATTICE2:th.79">LATTICE2:79</a></i>;<br/><b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p'</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_1"/>
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 <b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E1:76_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:76"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</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">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p</font></span><br/><b>consider </b><font color="Maroon" title="c3">q'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E2:76_2"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c3">q'</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> 
 <b>by </b><i><a class="txt" href="#E3:76"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/><b>take </b><font color="Maroon" title="c4">r</font> =  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c3">q'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p</font></span><br/><a NAME="E3:76_2"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> &amp; <font color="Maroon" title="c3">q'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:76"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="#E2:76_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="lattice2.html#T63" title="LATTICE2:th.63">LATTICE2:63</a>, <a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/><a NAME="E4:76_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c3">q'</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c3">q'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="lattice2.html#T78" title="LATTICE2:th.78">LATTICE2:78</a>, <a class="ref" href="lattice2.html#T79" title="LATTICE2:th.79">LATTICE2:79</a></i>;<br/><b>hence </b><a NAME="E5:76_2"/>
<font color="Maroon" title="c4">r</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76"/>
<font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 <b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E3:76"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
