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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</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">L1</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> ) iff <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> )</span><br/>



<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> implies <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> implies ( <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/>
( <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a></span><br/>

<b>hence </b><a NAME="E2:61_1"/>
( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> )
 <b>by </b><i><a class="ref" href="filter_1.html#T40" target="_self" title="FILTER_1:th.40">Th40</a>, <a class="ref" href="filter_1.html#T41" target="_self" title="FILTER_1:th.41">Th41</a></i>; <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="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="E2:61"/>
( <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> )</span><br/>

<b>hence </b><a NAME="E3:61"/>
( <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c1">L1</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> )
 <b>by </b><i><a class="ref" href="filter_1.html#T40" target="_self" title="FILTER_1:th.40">Th40</a>, <a class="ref" href="filter_1.html#T41" target="_self" title="FILTER_1:th.41">Th41</a></i>; <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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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