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<div class="add">

<b>set </b><font color="Maroon" title="c1">R</font> =  <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> ;<br/>
<a NAME="E1:29"/><i><font color="Green" title="E22">A1</font></i>: 
1 <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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</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>  <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<a NAME="E3:29"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</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 <span class="p1">(<span class="default"><a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="msualg_7.html#D4" target="_self" title="MSUALG_7:def.4">Def4</a></i>;<br/>
<b>set </b><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="c2">A</font>;<br/>
<b>set </b><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="c2">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">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="c2">A</font>, <font color="Maroon" title="c7">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="c2">A</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">L</font> =  <a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 as  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="ref" href="msualg_7.html#T21" target="_self" title="MSUALG_7:th.21">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">B</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E6:29"/><i><font color="Green" title="E24">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_1"/>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="#E1:29_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> </span><br/><div class="add"><a NAME="E2:29_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:29_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> </span><br/><div class="add"><a NAME="E2:29_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E4:29_1_1_1_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><a NAME="E5:29_1_1_1_1_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:29_1_1_1_1_2"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:29_1_1_1_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:29_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<b>assume </b><a NAME="E1:29_1_2"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_1_1"/>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E1:29_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/><div class="add"><a NAME="E2:29_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:29_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_2_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E3:29_1_2_1_1_2"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><a NAME="E4:29_1_2_1_1_2"/>
( <font color="Maroon" title="c11">y</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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E3:29_1_2_1_1_2"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/><a NAME="E5:29_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:29_1_2_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="#E3:29_1_2_1_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">B</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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>  ;<br/>
<div><i><font color="Green" title="E25">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> implies <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1_1"/>
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E1:29_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/><div class="add"><a NAME="E2:29_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E3:29_2_1_1_2"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><a NAME="E4:29_2_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="#E3:29_2_1_1_2"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:29_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:29_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:29_2_1_1_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E7:29_2_1_1_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:29_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:29"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c10">B</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"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">ma</font> = <font color="Maroon" title="c6">Ma</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c10">B</font>, <font color="Maroon" title="c12">mi</font> = <font color="Maroon" title="c7">Mi</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c10">B</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="c10">B</font>,<font color="Maroon" title="c10">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E12:29"/><i><font color="Green" title="E28">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">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="c10">B</font>,<font color="Maroon" title="c10">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">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="c10">B</font>,<font color="Maroon" title="c10">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">ma</font> implies <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> )</span><br/><b>assume </b><a NAME="E1:29_3"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><b>then consider </b><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_3"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c13">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="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:29_3"/><i><font color="Green" title="E31">A16</font></i>: 
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1"/>
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_1"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_3_1_1_1"/><b>then </b><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1_1_1_1"/>
( <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> = 
<font color="Maroon" title="c6">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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">x1</font>,<font color="Maroon" title="c15">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,1
<b>by </b><i><a class="txt" href="#E2:29_3_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="#E2:29_3_1_1_1_1_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></i>
<br/>.= 
1

;<br/><a NAME="E4:29_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:29_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E3:29_3_1_1_1_1_1_2"/><i><font color="Green" title="E33">A19</font></i>: 
( <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">y2</font> &amp; <font color="Maroon" title="c16">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> = 
<font color="Maroon" title="c6">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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">x1</font>,<font color="Maroon" title="c15">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> 1,<font color="Maroon" title="c16">y2</font>
<b>by </b><i><a class="txt" href="#E2:29_3_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="#E3:29_3_1_1_1_1_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E5:29_3_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><a NAME="E6:29_3_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E3:29_3_1_1_1_1_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_3_1_1_1"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_2"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">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> <b> such that </b><br/><a NAME="E3:29_3_1_1_2"/><i><font color="Green" title="E32">A20</font></i>: 
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c16">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1_2_1_1"/>
( <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> = 
<font color="Maroon" title="c6">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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">x1</font>,<font color="Maroon" title="c15">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c16">y1</font>,1
<b>by </b><i><a class="txt" href="#E3:29_3_1_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E2:29_3_1_1_2_1_1_1"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E4:29_3_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> or <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><a NAME="E5:29_3_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  or <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="#E3:29_3_1_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:29_3_1_1_2_1_1_2"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y2</font> &amp; <font color="Maroon" title="c17">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> = 
<font color="Maroon" title="c6">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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">x1</font>,<font color="Maroon" title="c15">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E3:29_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c17">y2</font>
<b>by </b><i><a class="txt" href="#E3:29_3_1_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E3:29_3_1_1_2_1_1_2"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E5:29_3_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> or <font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/><a NAME="E6:29_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:29_3_1_1_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E3:29_3_1_1_2_1_1_2"><i><font color="Green" title="E33">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E7:29_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:29_3_1_1_2"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:29_3"/>
<font color="Maroon" title="c11">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">ma</font> = <font color="Maroon" title="c11">ma</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c10">B</font> <b>by </b><i><a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">mi</font> implies <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> )</span><br/><b>assume </b><a NAME="E1:29_4"/><i><font color="Green" title="E29">A23</font></i>: 
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">mi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><b>then consider </b><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_4"/><i><font color="Green" title="E30">A24</font></i>: 
<font color="Maroon" title="c14">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="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:29_4"/><i><font color="Green" title="E31">A25</font></i>: 
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> )
 <b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1"/>
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_4"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_1"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_4_1_1_1"/><b>then </b><i><font color="Green" title="E32">A26</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1_1_1_1"/>
( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_4"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> = 
<font color="Maroon" title="c7">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c15">x1</font>,<font color="Maroon" title="c16">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 1,1
<b>by </b><i><a class="txt" href="#E2:29_4_1_1_1"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="#E2:29_4_1_1_1_1_1_1"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
<br/>.= 
1

;<br/><a NAME="E4:29_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:29_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:29_4_1_1_1_1_1_2"/><i><font color="Green" title="E33">A28</font></i>: 
( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y2</font> &amp; <font color="Maroon" title="c17">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> = 
<font color="Maroon" title="c7">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c15">x1</font>,<font color="Maroon" title="c16">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 1,<font color="Maroon" title="c17">y2</font>
<b>by </b><i><a class="txt" href="#E2:29_4_1_1_1"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="#E3:29_4_1_1_1_1_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E5:29_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><a NAME="E6:29_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E3:29_4_1_1_1_1_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_4_1_1_1"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_2"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:29_4_1_1_2"/><i><font color="Green" title="E32">A29</font></i>: 
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1_2_1_1"/>
( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E4:29_4"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><a NAME="E2:29_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> = 
<font color="Maroon" title="c7">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c15">x1</font>,<font color="Maroon" title="c16">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c17">y1</font>,1
<b>by </b><i><a class="txt" href="#E3:29_4_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E2:29_4_1_1_2_1_1_1"><i><font color="Green" title="E33">A30</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E4:29_4_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> or <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><a NAME="E5:29_4_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  or <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="#E3:29_4_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">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> <b> such that </b><br/><a NAME="E3:29_4_1_1_2_1_1_2"/><i><font color="Green" title="E33">A31</font></i>: 
( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">y2</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> = 
<font color="Maroon" title="c7">Mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c15">x1</font>,<font color="Maroon" title="c16">x2</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E1:29_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="#E3:29_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c17">y1</font>,<font color="Maroon" title="c18">y2</font>
<b>by </b><i><a class="txt" href="#E3:29_4_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E3:29_4_1_1_2_1_1_2"><i><font color="Green" title="E33">A31</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E5:29_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> or <font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><a NAME="E6:29_4_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:29_4_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E3:29_4_1_1_2_1_1_2"><i><font color="Green" title="E33">A31</font></i></a></i>;<br/><b>hence </b><a NAME="E7:29_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:29_4_1_1_2"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:29_4"/>
<font color="Maroon" title="c12">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">mi</font> = <font color="Maroon" title="c12">mi</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c10">B</font> <b>by </b><i><a class="txt" href="#E12:29"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">L'</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c14">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><i><font color="Green" title="E29">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x1</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="E1:29_5"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><a NAME="E2:29_5"/><b>then </b>
<font color="Maroon" title="c16">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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E4:29_5"/><i><font color="Green" title="E30">A33</font></i>: 
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:29_5"/>
<font color="Maroon" title="c16">x1</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> 
 <b>by </b><i><a class="txt" href="#E4:29_5"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="real_lat.html#D4" title="REAL_LAT:def.4">REAL_LAT:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E30">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_6_1"/>
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1_1"><b>suppose </b></a><a NAME="E1:29_6_1_1"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font></span><br/><div class="add"><a NAME="E2:29_6_1_1"/><b>then </b>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_6_1_1"/>
<font color="Maroon" title="c16">x1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>
 <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1_2"><b>suppose </b></a><a NAME="E1:29_6_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E3:29_6_1_2"/><i><font color="Green" title="E31">A35</font></i>: 
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><a NAME="E4:29_6_1_2"/>
<font color="Maroon" title="c17">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="#E3:29_6_1_2"><i><font color="Green" title="E31">A35</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:29_6_1_2"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:29_6_1_2"/><b>then </b>
<font color="Maroon" title="c16">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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:29_6_1_2"><i><font color="Green" title="E31">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E7:29_6_1_2"/>
<font color="Maroon" title="c16">x1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E31">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">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="c16">a</font>,<font color="Maroon" title="c17">b</font> &amp; <font color="Maroon" title="c16">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">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="c16">a</font>,<font color="Maroon" title="c17">b</font> )</span><br/><a NAME="E1:29_7"/><i><font color="Green" title="E32">A37</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c10">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>thus </b><font color="Maroon" title="c16">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">b</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c15">L'</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <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="c16">a</font>,<font color="Maroon" title="c17">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c16">a</font>,<font color="Maroon" title="c17">b</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E1:29_7"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">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="c16">a</font>,<font color="Maroon" title="c17">b</font></span><br/><b>thus </b><font color="Maroon" title="c16">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">b</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c15">L'</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <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="c16">a</font>,<font color="Maroon" title="c17">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c16">a</font>,<font color="Maroon" title="c17">b</font>
<b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="#E1:29_7"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E32">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">p</font></span><br/><b>reconsider </b><font color="Maroon" title="c18">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q'</font> = <font color="Maroon" title="c17">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">p'</font> = <font color="Maroon" title="c18">p'</font>, <font color="Maroon" title="c21">q'</font> = <font color="Maroon" title="c19">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">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="c16">p</font>,<font color="Maroon" title="c17">q</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c21">q'</font>,<font color="Maroon" title="c20">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T8" title="REAL_LAT:th.8">REAL_LAT:8</a></i>
<br/>.= 
<font color="Maroon" title="c17">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c16">p</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E33">A39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c18">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c18">r</font></span><br/><b>reconsider </b><font color="Maroon" title="c19">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c20">q'</font> = <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c21">r'</font> = <font color="Maroon" title="c18">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">p'</font> = <font color="Maroon" title="c19">p'</font>, <font color="Maroon" title="c23">q'</font> = <font color="Maroon" title="c20">q'</font>, <font color="Maroon" title="c24">r'</font> = <font color="Maroon" title="c21">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c18">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="c16">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c18">r</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c16">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="c17">q</font>,<font color="Maroon" title="c18">r</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c22">p'</font>,<font color="Maroon" title="c23">q'</font></span>)</span>,<font color="Maroon" title="c24">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T10" title="REAL_LAT:th.10">REAL_LAT:10</a></i>
<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="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font></span>)</span>,<font color="Maroon" title="c18">r</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c18">r</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E34">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font></span><br/><b>reconsider </b><font color="Maroon" title="c18">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q'</font> = <font color="Maroon" title="c17">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">p'</font> = <font color="Maroon" title="c18">p'</font>, <font color="Maroon" title="c21">q'</font> = <font color="Maroon" title="c19">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">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="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span>,<font color="Maroon" title="c17">q</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c20">p'</font>,<font color="Maroon" title="c21">q'</font></span>)</span>,<font color="Maroon" title="c21">q'</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">q</font>
<b>by </b><i><a class="ref" href="real_lat.html#T12" title="REAL_LAT:th.12">REAL_LAT:12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E35">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">p</font></span><br/><b>reconsider </b><font color="Maroon" title="c18">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q'</font> = <font color="Maroon" title="c17">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">p'</font> = <font color="Maroon" title="c18">p'</font>, <font color="Maroon" title="c21">q'</font> = <font color="Maroon" title="c19">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">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="c16">p</font>,<font color="Maroon" title="c17">q</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c21">q'</font>,<font color="Maroon" title="c20">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T9" title="REAL_LAT:th.9">REAL_LAT:9</a></i>
<br/>.= 
<font color="Maroon" title="c17">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c16">p</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E36">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c18">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c18">r</font></span><br/><b>reconsider </b><font color="Maroon" title="c19">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c20">q'</font> = <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c21">r'</font> = <font color="Maroon" title="c18">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">p'</font> = <font color="Maroon" title="c19">p'</font>, <font color="Maroon" title="c23">q'</font> = <font color="Maroon" title="c20">q'</font>, <font color="Maroon" title="c24">r'</font> = <font color="Maroon" title="c21">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c18">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="c16">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">q</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c18">r</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c16">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="c17">q</font>,<font color="Maroon" title="c18">r</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c22">p'</font>,<font color="Maroon" title="c23">q'</font></span>)</span>,<font color="Maroon" title="c24">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T11" title="REAL_LAT:th.11">REAL_LAT:11</a></i>
<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="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span>,<font color="Maroon" title="c18">r</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c18">r</font>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c17">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font></span><br/><b>reconsider </b><font color="Maroon" title="c18">p'</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q'</font> = <font color="Maroon" title="c17">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">p'</font> = <font color="Maroon" title="c18">p'</font>, <font color="Maroon" title="c21">q'</font> = <font color="Maroon" title="c19">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>  <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E18:29"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c16">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">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="c16">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c17">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<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="c20">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="c20">p'</font>,<font color="Maroon" title="c21">q'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A36</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">p</font>
<b>by </b><i><a class="ref" href="real_lat.html#T13" title="REAL_LAT:th.13">REAL_LAT:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E27:29"/><b>then </b>
( <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> &amp; <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> &amp; <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp; <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp; <font color="Maroon" title="c15">L'</font> is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="#E21:29"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="#E22:29"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="#E23:29"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="#E24:29"><i><font color="Green" title="E35">A41</font></i></a>, <a class="txt" href="#E25:29"><i><font color="Green" title="E36">A42</font></i></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></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">L'</font> = <font color="Maroon" title="c15">L'</font> as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">L'</font> = <font color="Maroon" title="c16">L'</font> as    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of  <a href="msualg_7.html#K1" title="MSUALG_7:func.1">RealSubLatt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E9:29"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="nat_lat.html#D16" title="NAT_LAT:def.16">NAT_LAT:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">L'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">L'</font> is <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a> &amp; not <font color="Maroon" title="c17">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c17">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">L'</font></span><br/><b>thus </b><a NAME="E1:29_14"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">L'</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_14_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>
<a NAME="E2:29_14_1"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">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> <b> such that </b><br/><a NAME="E4:29_14_1"/><i><font color="Green" title="E37">A43</font></i>: 
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">y</font> &amp; <font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:29_14_1"/><i><font color="Green" title="E38">A44</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>assume </b><a NAME="E6:29_14_1"/><i><font color="Green" title="E39">A45</font></i>: 
not  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">L'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:29_14_1"/><b>then </b>
( not  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:29_14_1"/><b>then </b><i><font color="Green" title="E40">A46</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">y</font>
 
<b>by </b><i><a class="txt" href="#E4:29_14_1"><i><font color="Green" title="E37">A43</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">z'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> implies <font color="Maroon" title="c20">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>assume </b><a NAME="E1:29_14_1_1"/><i><font color="Green" title="E41">A47</font></i>: 
<font color="Maroon" title="c20">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>then reconsider </b><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c20">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">L'</font> ;<br/><b>reconsider </b><font color="Maroon" title="c22">z</font> = <font color="Maroon" title="c21">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E8:29"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">z1</font> = <font color="Maroon" title="c22">z</font> as    <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>  <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:29_14_1_1"/><i><font color="Green" title="E42">A48</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">z</font>
 <b>by </b><i><a class="txt" href="#E5:29_14_1"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="#E1:29_14_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/> <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c23">z1</font>,<font color="Maroon" title="c19">y</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="c22">z</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:29_14_1"><i><font color="Green" title="E37">A43</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <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="c22">z</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">z</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c22">z</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c19">y</font>
<b>by </b><i><a class="txt" href="#E4:29_14_1"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="#E5:29_14_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
;<br/><a NAME="E7:29_14_1_1"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">z1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E8:29_14_1_1"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="txt" href="#E8:29_14_1"><i><font color="Green" title="E40">A46</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E9:29_14_1_1"/><b>then </b>
for <font color="Olive" title="b1">v</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>   holds <br/>(  not <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> or  not <font color="Olive" title="b1">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:29_14_1_1"/><b>then </b>
not <font color="Maroon" title="c23">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E11:29_14_1_1"/>
<font color="Maroon" title="c20">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:29_14_1"/><b>then </b><i><font color="Green" title="E41">A49</font></i>: 
<font color="Maroon" title="c18">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:29_14_1"/>
1 <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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">w</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_14_1_2_1"/>
( <font color="Maroon" title="c18">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c18">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="#E10:29_14_1"><i><font color="Green" title="E41">A49</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_14_1_2_1_1"/>
<font color="Maroon" title="c18">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_14_1_2_1_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> holds <br/><font color="Maroon" title="c20">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 ;<br/><a NAME="E3:29_14_1_2_1_1"/><b>then </b><i><font color="Green" title="E42">A50</font></i>: 
<font color="Maroon" title="c20">w</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">r1</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font> implies <font color="Maroon" title="c21">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c20">w</font> )</span><br/><a NAME="E1:29_14_1_2_1_1_1"/>
<font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><a NAME="E2:29_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">r1</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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">e</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> <b> such that </b><br/><a NAME="E4:29_14_1_2_1_1_1"/><i><font color="Green" title="E43">A51</font></i>: 
( <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">e</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">e</font> &amp; <font color="Maroon" title="c22">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_14_1_2_1_1_1"/>
<font color="Maroon" title="c21">r1</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c20">w</font></span><br/><font color="Maroon" title="c21">r1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c20">w</font> = 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">r1</font>,<font color="Maroon" title="c20">w</font>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<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="c21">r1</font>,<font color="Maroon" title="c20">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c21">r1</font>,<font color="Maroon" title="c20">w</font>

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c22">e</font>,1
<b>by </b><i><a class="txt" href="#E4:29_14_1_2_1_1_1"><i><font color="Green" title="E43">A51</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c21">r1</font>
<b>by </b><i><a class="txt" href="#E4:29_14_1_2_1_1_1"><i><font color="Green" title="E43">A51</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E7:29_14_1_2_1_1_1"/>
<font color="Maroon" title="c21">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c20">w</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:29_14_1_2_1_1"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font>
 <b>by </b><i><a class="txt" href="#E3:29_14_1_2_1_1"><i><font color="Green" title="E42">A50</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/><a NAME="E6:29_14_1_2_1_1"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_14_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E6:29_14_1"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_14_1_2_1_2"/>
<font color="Maroon" title="c18">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:29_14_1_2_1_2"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T43" title="LATTICE3:th.43">LATTICE3:43</a></i>;<br/><a NAME="E3:29_14_1_2_1_2"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_14_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E6:29_14_1"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:29_14_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E31:29"/>
<font color="Maroon" title="c17">L'</font> is <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a>
 <b>by </b><i><a class="ref" href="msualg_7.html#D2" target="_self" title="MSUALG_7:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c17">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c18">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> ;<br/><a NAME="E1:29_15"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c17">L'</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c20">X</font> = <font color="Maroon" title="c19">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c20">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">L'</font></span><br/><a NAME="E3:29_15"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">z</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font> implies <font color="Maroon" title="c22">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c21">z</font> )</span><br/><a NAME="E1:29_15_1"/>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><a NAME="E2:29_15_1"/><b>then </b>
<font color="Maroon" title="c22">q</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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">q'</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> <b> such that </b><br/><a NAME="E4:29_15_1"/><i><font color="Green" title="E37">A52</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q'</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">q'</font> &amp; <font color="Maroon" title="c23">q'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_15_1"/>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c21">z</font></span><br/><b>then consider </b><font color="Maroon" title="c24">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> <b> such that </b><br/><a NAME="E7:29_15_1"/><i><font color="Green" title="E38">A53</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">y</font> &amp; <font color="Maroon" title="c24">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><font color="Maroon" title="c22">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c21">z</font> = 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c21">z</font>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<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="c22">q</font>,<font color="Maroon" title="c21">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c22">q</font>,<font color="Maroon" title="c21">z</font>

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c23">q'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E4:29_15_1"><i><font color="Green" title="E37">A52</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c22">q</font>
<b>by </b><i><a class="txt" href="#E4:29_15_1"><i><font color="Green" title="E37">A52</font></i></a>, <a class="txt" href="#E7:29_15_1"><i><font color="Green" title="E38">A53</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E9:29_15_1"/>
<font color="Maroon" title="c22">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:29_15"/><b>then </b><i><font color="Green" title="E37">A54</font></i>: 
<font color="Maroon" title="c20">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c22">b</font> implies <font color="Maroon" title="c21">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font> )</span><br/><a NAME="E1:29_15_2"/>
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><a NAME="E2:29_15_2"/><b>then </b>
<font color="Maroon" title="c22">b</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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">b'</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> <b> such that </b><br/><a NAME="E4:29_15_2"/><i><font color="Green" title="E38">A55</font></i>: 
( <font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b'</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">b'</font> &amp; <font color="Maroon" title="c23">b'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_15_2"/><i><font color="Green" title="E39">A56</font></i>: 
<font color="Maroon" title="c20">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c22">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font></span><br/><a NAME="E6:29_15_2"/><i><font color="Green" title="E40">A57</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_15_2_1"/><i><font color="Green" title="E41">A58</font></i>: 
<font color="Maroon" title="c23">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:29_15_2_1"/><b>then </b>
<font color="Maroon" title="c23">b'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:29_15_2_1"/><b>then </b><i><font color="Green" title="E42">A59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">b'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E4:29_15_2_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/>
<a NAME="E6:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">c</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">L</font> <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E8:29_15_2_1"/>
<font color="Maroon" title="c23">b'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:29_15_2_1"><i><font color="Green" title="E41">A58</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E10:29_15_2_1"/><b>then </b><i><font color="Green" title="E43">A60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font>
 
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a></i>;<br/>
<a NAME="E11:29_15_2_1"/>
<font color="Maroon" title="c23">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font>
 
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="txt" href="#E1:29_15_2_1"><i><font color="Green" title="E41">A58</font></i></a></i>;<br/>
<a NAME="E12:29_15_2_1"/><b>then </b>
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c20">X</font>,<font color="Maroon" title="c8">L</font>
 
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="txt" href="#E5:29_15_2"><i><font color="Green" title="E39">A56</font></i></a>, <a class="ref" href="lattice3.html#T41" title="LATTICE3:th.41">LATTICE3:41</a></i>;<br/>
<a NAME="E13:29_15_2_1"/><b>then </b>
<font color="Maroon" title="c24">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font>
 
<b>by </b><i><a class="txt" href="#E10:29_15_2_1"><i><font color="Green" title="E43">A60</font></i></a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 = 
<font color="Maroon" title="c24">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c22">b</font>
<b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">c</font>,<font color="Maroon" title="c22">b</font>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<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="c24">c</font>,<font color="Maroon" title="c22">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c24">c</font>,<font color="Maroon" title="c22">b</font>

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<font color="Maroon" title="c23">b'</font>
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E15:29_15_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:29_15_2_1"><i><font color="Green" title="E42">A59</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><font color="Maroon" title="c21">z</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c22">b</font> = 
<span class="p1">(<span class="default"><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="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">z</font>,<font color="Maroon" title="c22">b</font>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<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="c21">z</font>,<font color="Maroon" title="c22">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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="c21">z</font>,<font color="Maroon" title="c22">b</font>

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<font color="Maroon" title="c23">b'</font>
<b>by </b><i><a class="txt" href="#E4:29_15_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c21">z</font>
<b>by </b><i><a class="txt" href="#E6:29_15_2"><i><font color="Green" title="E40">A57</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
;<br/><b>hence </b><a NAME="E8:29_15_2"/>
<font color="Maroon" title="c21">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E8:29_15"/><b>then </b><i><font color="Green" title="E38">A61</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c20">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="#E6:29_15"><i><font color="Green" title="E37">A54</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/><a NAME="E9:29_15"/><i><font color="Green" title="E39">A62</font></i>: 
not 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:29_15"/>
for <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>   holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> or  not <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/><a NAME="E11:29_15"/><b>then </b>
not 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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>  : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E12:29_15"/>
not  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c20">X</font>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">L'</font>
 <b>by </b><i><a class="txt" href="#E6:29"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E8:29_15"><i><font color="Green" title="E38">A61</font></i></a>, <a class="txt" href="#E9:29_15"><i><font color="Green" title="E39">A62</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E33:29"/>
not <font color="Maroon" title="c17">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a>
 <b>by </b><i><a class="ref" href="msualg_7.html#D3" target="_self" title="MSUALG_7:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
