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<span class="kw">set </span><font color="Maroon" title="c2">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #);<br/>
<a NAME="E1:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T22" target="_self" title="REAL_LAT:th.22">Th22</a></span>;<br/>
<a NAME="E2:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></span>;<br/>
<a NAME="E3:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T26" target="_self" title="REAL_LAT:th.26">Th26</a></span>;<br/>
<a NAME="E4:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a></span>;<br/>
<a NAME="E5:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T28" target="_self" title="REAL_LAT:th.28">Th28</a></span>;<br/>
<a NAME="E6:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<a NAME="E7:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T23" target="_self" title="REAL_LAT:th.23">Th23</a></span>;<br/>
<a NAME="E8:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>;<br/>
<a NAME="E9:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T21" target="_self" title="REAL_LAT:th.21">Th21</a></span>;<br/>
<a NAME="E10:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a></span>;<br/>
<a NAME="E11:43_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) holds  <font color="Olive" title="b1">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T20" target="_self" title="REAL_LAT:th.20">Th20</a></span>;<br/>
<a NAME="E12:43_1_1"/><span class="kw">then </span>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_1_1"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> #) is   <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E4:43_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E10:43_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E8:43_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E6:43_1_1"><span class="lab"><font color="Green" title="E35">A3</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"> verum</span><br/>


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