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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c1">L1</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:61_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c1">L1</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:61_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c2">L2</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 <span class="kw">and </span><br/><a NAME="E4:61_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c2">L2</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E5:61_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E2:61_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <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></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E4:61"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <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></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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