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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>) )</span><br/>

<div><span class="lab"><font color="Green" title="E49">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">now </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 <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</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="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>) &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>) )</span><br/><a NAME="E1:86_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">C</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D17" target="_self" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:86_1_1"/>
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>) <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D21" target="_self" title="LATTICE3:def.21">Def21</a></span>;<br/><a NAME="E3:86_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:86_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</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="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="lattices.html#T8" title="LATTICES:th.8">LATTICES:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span><br/><span class="kw">thus </span><a NAME="E5:86_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:86_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="lattices.html#T8" title="LATTICES:th.8">LATTICES:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:86"/><span class="kw">then </span>
<font color="Maroon" title="c1">C</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:86"/>
( <font color="Maroon" title="c1">C</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c1">C</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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