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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="nat_lat.html#K3" title="NAT_LAT:func.3">Nat_Lattice</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">r</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c2">q</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c1">p</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">q</font> )</span><br/>





<b>set </b><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">r</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">q</font>;<br/>
<b>thus </b><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> = 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="ref" href="nat_lat.html#T65" target="_self" title="NAT_LAT:th.65">Th65</a></i>
<br/>.= 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="ref" href="nat_lat.html#T62" target="_self" title="NAT_LAT:th.62">Th62</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c2">q</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c1">p</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">q</font> )</span><br/>

<b>thus </b><i><font color="Green" title="E8">A1</font></i>: <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> = 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_lat.html#T62" target="_self" title="NAT_LAT:th.62">Th62</a></i>
<br/>.= 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c2">q</font>
<b>by </b><i><a class="ref" href="nat_lat.html#T65" target="_self" title="NAT_LAT:th.65">Th65</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c1">p</font> &amp; <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">q</font> )</span><br/>

<b>thus </b><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> = 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="nat_lat.html#T62" target="_self" title="NAT_LAT:th.62">Th62</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">q</font></span><br/>

<b>thus </b><a NAME="E4:15"/>
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="nat_lat.html#E2:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="nat_lat.html#T62" target="_self" title="NAT_LAT:th.62">Th62</a></i>; <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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