<?xml version="1.0"?>
<div class="add">

<a NAME="E1:25_1_1"/>
<font color="Maroon" title="c2">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E3:25_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Ma</font> = <a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c5">Mi</font> = <a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">A</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>;<br/>
<a NAME="E5:25_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Mi</font> implies <font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Mi</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="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:25_1_1_1"/>
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:25_1_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:25_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E7:25_1_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">and </span><br/><a NAME="E8:25_1_1_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y2</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> )
 ;<br/><a NAME="E9:25_1_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:25_1_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:25_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E12:25_1_1_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font>
 <span class="kw">and </span><br/><a NAME="E13:25_1_1_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> )
 ;<br/><font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:25_1_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c9">y2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E7:25_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
;<br/><a NAME="E15:25_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> or <font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><a NAME="E16:25_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E8:25_1_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E17:25_1_1_1"/>
<font color="Maroon" title="c5">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E21">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">Ma</font> implies <font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">Ma</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="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1_2"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:25_1_1_2"/>
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E3:25_1_1_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:25_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E7:25_1_1_2"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">and </span><br/><a NAME="E8:25_1_1_2"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y2</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> )
 ;<br/><a NAME="E9:25_1_1_2"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E3:25_1_1_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:25_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E12:25_1_1_2"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font>
 <span class="kw">and </span><br/><a NAME="E13:25_1_1_2"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> )
 ;<br/><font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E3:25_1_1_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c9">y2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1_2"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E7:25_1_1_2"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></span>
;<br/><a NAME="E15:25_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> or <font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E16:25_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1_2"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="txt" href="#E8:25_1_1_2"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E17:25_1_1_2"/>
<font color="Maroon" title="c4">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</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/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Mi</font> = <font color="Maroon" title="c5">Mi</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<a NAME="E9:25_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">Ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Ma</font> = <font color="Maroon" title="c4">Ma</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">R</font> =  <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">L9</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">Ma</font>,<font color="Maroon" title="c6">Mi</font> #);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">Ma</font>,<font color="Maroon" title="c6">Mi</font> #) as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a>  ;<br/>
<div><span class="lab"><font color="Green" title="E22">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> &amp; <font color="Maroon" title="c11">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c11">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">b</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c10">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</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="c11">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c11">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">b</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c10">L</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E23">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</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="c11">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> </span><br/><a NAME="E1:25_1_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><a NAME="E2:25_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><a NAME="E3:25_1_1_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r2</font> )
 ;<br/><span class="kw">hence </span><a NAME="E4:25_1_1_4"/>
<font color="Maroon" title="c11">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#D4" title="REAL_LAT:def.4">REAL_LAT:def 4</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E24">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c11">p</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font> = 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">q9</font>,<font color="Maroon" title="c13">p9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T8" title="REAL_LAT:th.8">REAL_LAT:8</a></span>
<br/>.= 
<font color="Maroon" title="c12">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E25">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span> = 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">p9</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">p9</font>,<font color="Maroon" title="c14">q9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T13" title="REAL_LAT:th.13">REAL_LAT:13</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E26">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c11">p</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font> = 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">q9</font>,<font color="Maroon" title="c13">p9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T9" title="REAL_LAT:th.9">REAL_LAT:9</a></span>
<br/>.= 
<font color="Maroon" title="c12">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E27">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font> = 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">p9</font>,<font color="Maroon" title="c14">q9</font></span>)</span>,<font color="Maroon" title="c14">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T12" title="REAL_LAT:th.12">REAL_LAT:12</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E28">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">r</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c16">r9</font> = <font color="Maroon" title="c13">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">r</font></span>)</span> = 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">q9</font></span>)</span>,<font color="Maroon" title="c16">r9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T11" title="REAL_LAT:th.11">REAL_LAT:11</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span>,<font color="Maroon" title="c13">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c13">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c13">r</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c16">r9</font> = <font color="Maroon" title="c13">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c13">r</font></span>)</span> = 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c13">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">q9</font></span>)</span>,<font color="Maroon" title="c16">r9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T10" title="REAL_LAT:th.10">REAL_LAT:10</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span>,<font color="Maroon" title="c13">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c13">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E22">A17</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/></div><span class="kw">end;</span></div>
<a NAME="E20:25_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp; <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp; <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp; <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp; <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  &amp; <font color="Maroon" title="c10">L</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25_1_1"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E17:25_1_1"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E16:25_1_1"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E18:25_1_1"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E15:25_1_1"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">L9</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">Ma</font>,<font color="Maroon" title="c6">Mi</font> #) as   <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">L9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E22:25_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c11">L9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r1</font>,<font color="Maroon" title="c2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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/>


</div>
