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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  holds <br/> <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Olive" title="b1">F</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>; <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">L</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  implies  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><a NAME="E2:81"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:81"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/>
<a NAME="E5:81"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
 ex <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">F</font> st <br/>( <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> #) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D10" target="_self" title="FILTER_0:def.10">Def10</a></span>;<br/>
<a NAME="E6:81"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p9</font> = <font color="Maroon" title="c3">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="filter_0.html#T10" target="_self" title="FILTER_0:th.10">Th10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p9</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span> holds <br/> ( <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>; <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="c5">p9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> &amp; <font color="Maroon" title="c6">r</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">r9</font> = <font color="Maroon" title="c6">r</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81"><span class="lab"><font color="Green" title="E52">A3</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">p9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">r</font> = 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">r9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>
<br/>.= 
<font color="Maroon" title="c5">p9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>
; <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="c6">r</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font></span><br/>

<span class="kw">hence </span><a NAME="E10:81"/>
<font color="Maroon" title="c6">r</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font>
 ; <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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