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<a NAME="E1:25_1_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E3:25_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="supinf_1.html#K3">REAL</a> ,<a href="supinf_1.html#K3">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ma</font> = <a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font>, <font color="Maroon">Mi</font> = <a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E5:25_1_1"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ma</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Mi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ma</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:25_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:25_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E7:25_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><a NAME="E8:25_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E10:25_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E12:25_1_1_1"/><b>then </b>
( <font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><a NAME="E13:25_1_1_1"/><b>then </b>
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E14:25_1_1_1"/>
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Ma</font> = <font color="Maroon">Ma</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Mi</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:25_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:25_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E7:25_1_1_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><a NAME="E8:25_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E10:25_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E12:25_1_1_2"/><b>then </b>
( <font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><a NAME="E13:25_1_1_2"/><b>then </b>
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E14:25_1_1_2"/>
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Mi</font> = <font color="Maroon">Mi</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="real_lat.html#K3">Real_Lattice</a> ;<br/>
<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:25_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:25_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:25_1_1_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><b>thus </b><a NAME="E5:25_1_1_3"/>
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a> 
 <b>by </b><i>, <a class="ref" href="real_lat.html#D4">REAL_LAT:def 4</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">L'</font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">Ma</font>,<font color="Maroon">Mi</font> #);<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">Ma</font>,<font color="Maroon">Mi</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  ;<br/>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

;<br/></div><b>end;</b></div>
<div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q'</font>,<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T8">REAL_LAT:8</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T10">REAL_LAT:10</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">q'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="real_lat.html#T12">REAL_LAT:12</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q'</font>,<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T9">REAL_LAT:9</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T11">REAL_LAT:11</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="real_lat.html#T13">REAL_LAT:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E19:25_1_1"/><b>then </b>
( <font color="Maroon">L</font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V7">meet-associative</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i>, , , , , <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L'</font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">Ma</font>,<font color="Maroon">Mi</font> #) as  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">L'</font>
;<br/>

<b>thus </b><a NAME="E21:25_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a> <a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a> <a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>


</div>
