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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_lat.html#K7" target="_self">RealFunc_Lattice</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_lat.html#K7" target="_self">RealFunc_Lattice</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_lat.html#K7" target="_self">RealFunc_Lattice</a> <font color="Maroon">A</font></span>)</span>;<br/>







<b>set </b><font color="Maroon">s</font> = <font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font>;<br/>
<b>thus </b><i><font color="Green">E25</font></i>: <span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>,<font color="Maroon">p</font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="real_lat.html#E6"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="real_lat.html#E6"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">r</font>

;<br/>

<b>thus </b><i><font color="Green">E26</font></i>: <span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> = 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="real_lat.html#E6"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font>

;<br/>

<b>thus </b><a NAME="E5:41"/>
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="real_lat.html#E7"><i><font color="Green">Lemma34</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:41"/>
<span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" target="_self">minfuncreal</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="real_lat.html#E8"><i><font color="Green">Lemma36</font></i></a>, <a class="txt" href="real_lat.html#E7"><i><font color="Green">Lemma34</font></i></a></i>;<br/>


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