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

<span class="kw">set </span><font color="Maroon" title="c1">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/>
<span class="kw">set </span><font color="Maroon" title="c2">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/>
<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> ;<br/>
<a NAME="E1:28"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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="c4">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/>

<span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="28_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_1_1_1"/>
( <font color="Maroon" title="c4">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="c4">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1_1"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="E2:28_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">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="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E3:28_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">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="c4">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_1_1_1_1"/>
<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1_2"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="E2:28_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E4:28_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E5:28_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
(  <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="c5">y</font> &amp; <font color="Maroon" title="c5">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="E6:28_1_1_1_2"/>
<font color="Maroon" title="c5">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:28_1_1_1_2"/>
<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_1_1_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:28_1"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:28"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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="c4">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/>

<span class="kw">assume </span><a NAME="E1:28_2"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="28_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_2_1_1"/>
( <font color="Maroon" title="c4">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="c4">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_1"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="E2:28_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:28_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_2_1_1_1"/>
<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_2"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E3:28_2_1_1_2"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E4:28_2_1_1_2"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
(  <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="c5">y</font> &amp; <font color="Maroon" title="c5">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="E5:28_2_1_1_2"/>
( <font color="Maroon" title="c5">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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2_1_1_2"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>;<br/><a NAME="E6:28_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">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="c5">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> )
 <span class="kw">by</span> <span class="lab"><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></span>;<br/><span class="kw">hence </span><a NAME="E7:28_2_1_1_2"/>
<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:28_2"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E4:28"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">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="E1:28_3"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><a NAME="E2:28_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><a NAME="E3:28_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">y1</font> &amp; <font color="Olive" title="b1">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><span class="kw">hence </span><a NAME="E4:28_3"/>
<font color="Maroon" title="c5">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 class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#D3" title="REAL_LAT:def.3">REAL_LAT:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c5">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/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D4" target="_self" title="MSUALG_7:def.4">Def4</a></span>;<br/>
<span class="kw">set </span><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="c4">A</font>;<br/>
<span class="kw">set </span><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="c4">A</font>;<br/>
<a NAME="E9:28"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D4" target="_self" title="MSUALG_7:def.4">Def4</a></span>;<br/>
<a NAME="E10:28"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D4" target="_self" title="MSUALG_7:def.4">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">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="c4">A</font>, <font color="Maroon" title="c9">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="c4">A</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D4" target="_self" title="MSUALG_7:def.4">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E29">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> implies <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_4"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_4_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:28_4"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_4_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/><div class="add"><a NAME="E2:28_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_4_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_4_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:28_4_1_1_2"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E4:28_4_1_1_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<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="E5:28_4_1_1_2"/>
<font color="Maroon" title="c11">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_4_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E6:28_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:28_4_1_1_2"/><span class="kw">then </span>
<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#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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_4_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_4_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:28_4"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E14:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c5">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="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">ma</font> = <font color="Maroon" title="c8">Ma</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c11">mi</font> = <font color="Maroon" title="c9">Mi</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c5">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="c5">B</font>,<font color="Maroon" title="c5">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<a NAME="E16:28"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">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="c5">B</font>,<font color="Maroon" title="c5">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">ma</font> implies <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_5"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:28_5"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c12">x9</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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:28_5"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E5:28_5"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_5_1_1"/>
( <font color="Maroon" title="c13">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="c13">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E5:28_5"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_1"/>
<font color="Maroon" title="c13">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_5_1_1_1_1_1"/>
( <font color="Maroon" title="c14">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="c14">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:28_5"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_5_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c8">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (1,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_5_1_1_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E2:28_5_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></span>
<br/>.= 
1

;<br/><a NAME="E4:28_5_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:28_5_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_1_1_1_2"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:28_5_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <span class="kw">and </span><br/><a NAME="E4:28_5_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
(  <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="c15">y2</font> &amp; <font color="Maroon" title="c15">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="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c8">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (1,<font color="Maroon" title="c15">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_5_1_1_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E3:28_5_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></span>
;<br/><a NAME="E6:28_5_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E7:28_5_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_5_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_5_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:28_5_1_1_1"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_2"/>
<font color="Maroon" title="c13">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:28_5_1_1_2"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font>
 <span class="kw">and </span><br/><a NAME="E4:28_5_1_1_2"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
(  <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="c15">y1</font> &amp; <font color="Maroon" title="c15">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="28_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_5_1_1_2_1_1"/>
( <font color="Maroon" title="c14">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="c14">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:28_5"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_5_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c8">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c15">y1</font>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_5_1_1_2"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E2:28_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></span>
;<br/><a NAME="E4:28_5_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> or <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E5:28_5_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_5_1_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:28_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_5_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:28_5_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font>
 <span class="kw">and </span><br/><a NAME="E4:28_5_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
(  <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="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c8">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:28_5"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:28_5"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c16">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_5_1_1_2"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E3:28_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="real_lat.html#D2" title="REAL_LAT:def.2">REAL_LAT:def 2</a></span>
;<br/><a NAME="E6:28_5_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> or <font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E7:28_5_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_5_1_1_2"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E4:28_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:28_5_1_1_2"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:28_5"/>
<font color="Maroon" title="c10">ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:28"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">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="c5">B</font>,<font color="Maroon" title="c5">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">mi</font> implies <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_6"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">mi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:28_6"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
<font color="Maroon" title="c12">x9</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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E4:28_6"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E5:28_6"/><span class="lab"><font color="Green" title="E39">A40</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_6_1_1"/>
( <font color="Maroon" title="c13">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="c13">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E5:28_6"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_1"/>
<font color="Maroon" title="c13">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A41</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_6_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_6_1_1_1_1_1"/>
( <font color="Maroon" title="c14">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="c14">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:28_6"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_1_1_1_1"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_6_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A42</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c9">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (1,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_6_1_1_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E2:28_6_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
<br/>.= 
1

;<br/><a NAME="E4:28_6_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:28_6_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_1_1_1_2"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:28_6_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font>
 <span class="kw">and </span><br/><a NAME="E4:28_6_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
(  <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="c15">y2</font> &amp; <font color="Maroon" title="c15">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">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c9">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (1,<font color="Maroon" title="c15">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_6_1_1_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E3:28_6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
;<br/><a NAME="E6:28_6_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><a NAME="E7:28_6_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_6_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:28_6_1_1_1"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_2"/>
<font color="Maroon" title="c13">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:28_6_1_1_2"/><span class="lab"><font color="Green" title="E40">A45</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font>
 <span class="kw">and </span><br/><a NAME="E4:28_6_1_1_2"/><span class="lab"><font color="Green" title="E41">A46</font></span>: 
(  <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="c15">y1</font> &amp; <font color="Maroon" title="c15">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="28_6_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_6_1_1_2_1_1"/>
( <font color="Maroon" title="c14">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="c14">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:28_6"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><a NAME="E2:28_6_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A47</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c9">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c15">y1</font>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_6_1_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="txt" href="#E2:28_6_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A47</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
;<br/><a NAME="E4:28_6_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> or <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><a NAME="E5:28_6_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_6_1_1_2"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:28_6_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_6_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_6_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:28_6_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font>
 <span class="kw">and </span><br/><a NAME="E4:28_6_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A49</font></span>: 
(  <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">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> = 
<font color="Maroon" title="c9">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="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E1:28_6"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:28_6"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c16">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_6_1_1_2"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="txt" href="#E3:28_6_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
;<br/><a NAME="E6:28_6_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y1</font> or <font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><a NAME="E7:28_6_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_6_1_1_2"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="txt" href="#E4:28_6_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A49</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_6_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:28_6_1_1_2"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:28_6"/>
<font color="Maroon" title="c11">mi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c12">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#T20" target="_self" title="MSUALG_7:th.20">Th21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">mi</font> = <font color="Maroon" title="c11">mi</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c5">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E19:28"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">ma</font> = <font color="Maroon" title="c10">ma</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c5">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">L9</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c14">ma</font>,<font color="Maroon" title="c13">mi</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a>  ;<br/>
<div><span class="lab"><font color="Green" title="E36">A50</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E1:28_7"/><span class="lab"><font color="Green" title="E37">A51</font></span>: 
<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="c5">B</font>,<font color="Maroon" title="c5">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><span class="kw">thus </span><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">L9</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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<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="c4">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E1:28_7"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">thus </span><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">L9</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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<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="c4">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">a</font>,<font color="Maroon" title="c17">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E1:28_7"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E37">A52</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c5">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">L</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_8_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_8_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">L</font></span><br/><div class="add"><a NAME="E2:28_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_8_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="c12">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_8_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">L</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:28_8_1_2"/><span class="lab"><font color="Green" title="E38">A53</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E4:28_8_1_2"/><span class="lab"><font color="Green" title="E39">A54</font></span>: 
<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="E5:28_8_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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_8_1_2"><span class="lab"><font color="Green" title="E39">A54</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E6:28_8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:28_8_1_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_8_1_2"><span class="lab"><font color="Green" title="E38">A53</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:28_8_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="c12">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E38">A55</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">p9</font> = <font color="Maroon" title="c18">p9</font>, <font color="Maroon" title="c21">q9</font> = <font color="Maroon" title="c19">q9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c21">q9</font>,<font color="Maroon" title="c20">p9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T1" title="REAL_LAT:th.1">REAL_LAT:1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E39">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">p9</font> = <font color="Maroon" title="c18">p9</font>, <font color="Maroon" title="c21">q9</font> = <font color="Maroon" title="c19">q9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">p9</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">p9</font>,<font color="Maroon" title="c21">q9</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T6" title="REAL_LAT:th.6">REAL_LAT:6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E40">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">p9</font> = <font color="Maroon" title="c18">p9</font>, <font color="Maroon" title="c21">q9</font> = <font color="Maroon" title="c19">q9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c21">q9</font>,<font color="Maroon" title="c20">p9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T2" title="REAL_LAT:th.2">REAL_LAT:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E41">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_12"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c18">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c19">q9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">p9</font> = <font color="Maroon" title="c18">p9</font>, <font color="Maroon" title="c21">q9</font> = <font color="Maroon" title="c19">q9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">p9</font>,<font color="Maroon" title="c21">q9</font>)</span>)</span>,<font color="Maroon" title="c21">q9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T5" title="REAL_LAT:th.5">REAL_LAT:5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A59</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_13"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c19">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c20">q9</font> = <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c21">r9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">p9</font> = <font color="Maroon" title="c19">p9</font>, <font color="Maroon" title="c23">q9</font> = <font color="Maroon" title="c20">q9</font>, <font color="Maroon" title="c24">r9</font> = <font color="Maroon" title="c21">r9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c22">p9</font>,<font color="Maroon" title="c23">q9</font>)</span>)</span>,<font color="Maroon" title="c24">r9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T4" title="REAL_LAT:th.4">REAL_LAT:4</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><font color="Maroon" title="c19">p9</font> = <font color="Maroon" title="c16">p</font>, <font color="Maroon" title="c20">q9</font> = <font color="Maroon" title="c17">q</font>, <font color="Maroon" title="c21">r9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E37">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">p9</font> = <font color="Maroon" title="c19">p9</font>, <font color="Maroon" title="c23">q9</font> = <font color="Maroon" title="c20">q9</font>, <font color="Maroon" title="c24">r9</font> = <font color="Maroon" title="c21">r9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="real_lat.html#K3" title="REAL_LAT:func.3">Real_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c22">p9</font>,<font color="Maroon" title="c23">q9</font>)</span>)</span>,<font color="Maroon" title="c24">r9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_lat.html#T3" title="REAL_LAT:th.3">REAL_LAT:3</a></span>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E36">A50</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:28"/><span class="kw">then </span>
( <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp; <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp; <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp; <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp; <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  &amp; <font color="Maroon" title="c15">L9</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">L9</font> = <font color="Maroon" title="c15">L9</font> as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">L9</font> = <font color="Maroon" title="c16">L9</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) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="nat_lat.html#D12" title="NAT_LAT:def.12">NAT_LAT:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">L9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">L9</font> is  <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a>  &amp;  not <font color="Maroon" title="c17">L9</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="28_15"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c12">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">L9</font></span><br/><span class="kw">thus </span><a NAME="E1:28_15"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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">L9</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_15_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_15_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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">w</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E3:28_15_1"/><span class="lab"><font color="Green" title="E43">A60</font></span>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">L</font>) <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<a NAME="E4:28_15_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a></span>;<br/>
<a NAME="E5:28_15_1"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">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> <span class="kw"> such that </span><br/><a NAME="E7:28_15_1"/><span class="lab"><font color="Green" title="E44">A61</font></span>: 
<font color="Maroon" title="c20">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="c12">L</font>)
 <span class="kw">and </span><br/><a NAME="E8:28_15_1"/><span class="lab"><font color="Green" title="E45">A62</font></span>: 
 <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="c20">y</font>
 <span class="kw">and </span><br/><a NAME="E9:28_15_1"/>
<font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>
<span class="kw">assume </span><a NAME="E10:28_15_1"/><span class="lab"><font color="Green" title="E46">A63</font></span>: 
 not  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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">L9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E11:28_15_1"/><span class="kw">then </span>
 not  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:28_15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A64</font></span>: 
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="c20">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_15_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="txt" href="#E8:28_15_1"><span class="lab"><font color="Green" title="E45">A62</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_15_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">z9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> implies <font color="Maroon" title="c21">z9</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/><span class="kw">assume </span><a NAME="E1:28_15_1_1"/><span class="lab"><font color="Green" title="E48">A65</font></span>: 
<font color="Maroon" title="c21">z9</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">z9</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">z</font> = <font color="Maroon" title="c21">z9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">L9</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">z</font> = <font color="Maroon" title="c22">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><a NAME="E4:28_15_1_1"/><span class="lab"><font color="Green" title="E49">A66</font></span>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">L</font>) <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c23">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_15_1"><span class="lab"><font color="Green" title="E43">A60</font></span></a>, <a class="txt" href="#E1:28_15_1_1"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">z1</font> = <font color="Maroon" title="c23">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/> <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c24">z1</font>,<font color="Maroon" title="c20">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="c23">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="c12">L</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_15_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
<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="c4">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="c23">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="c12">L</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<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="c4">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c23">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="c12">L</font>)</span>)</span>)

<br/>.= 
<font color="Maroon" title="c23">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="c12">L</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_15_1"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="txt" href="#E4:28_15_1_1"><span class="lab"><font color="Green" title="E49">A66</font></span></a>, <a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>
;<br/><a NAME="E7:28_15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E8:28_15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">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="c24">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28_15_1"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E9:28_15_1_1"/><span class="kw">then </span>
 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="c24">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 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:28_15_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c24">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/><span class="kw">hence </span><a NAME="E11:28_15_1_1"/>
<font color="Maroon" title="c21">z9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:28_15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A67</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_15_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_15_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_15_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28_15_1"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_15_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_15_1_2_1_1"/><span class="lab"><font color="Green" title="E49">A68</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E50">A69</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_15_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c12">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c19">w</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_15_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c19">w</font></span><br/><a NAME="E2:28_15_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a></span>;<br/><a NAME="E3:28_15_1_2_1_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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> <span class="kw"> such that </span><br/><a NAME="E5:28_15_1_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
<font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">e</font>
 <span class="kw">and </span><br/><a NAME="E6:28_15_1_2_1_1_1"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E7:28_15_1_2_1_1_1"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
<font color="Maroon" title="c22">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><font color="Maroon" title="c21">r1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c19">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="c4">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="c19">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">r1</font>,<font color="Maroon" title="c19">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c21">r1</font>,<font color="Maroon" title="c19">w</font>)

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c22">e</font>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c21">r1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="txt" href="#E7:28_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:28_15_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="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:28_15_1_2_1_1"/>
 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="c12">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="c19">w</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_15_1_2_1_1"><span class="lab"><font color="Green" title="E49">A68</font></span></a></span>;<br/><a NAME="E4:28_15_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">w</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c18">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/><a NAME="E5:28_15_1_2_1_1"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_15_1_2_1_1"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/><a NAME="E6:28_15_1_2_1_1"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:28_15_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E10:28_15_1"><span class="lab"><font color="Green" title="E46">A63</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_15_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_15_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:28_15_1_2_1_2"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T42" title="LATTICE3:th.42">LATTICE3:42</a></span>;<br/><a NAME="E3:28_15_1_2_1_2"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c18">X</font>,<font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:28_15_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E10:28_15_1"><span class="lab"><font color="Green" title="E46">A63</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:28_15_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E36:28"/>
<font color="Maroon" title="c17">L9</font> is  <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D2" target="_self" title="MSUALG_7:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c17">L9</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="28_16"><span class="kw">now </span></a><div class="add"><a NAME="E1:28_16"/>
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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">set </span><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> ;<br/><a NAME="E3:28_16"/>
 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="E4:28_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A72</font></span>: 
(  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> &amp;  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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:28_16"/>
 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="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">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">L9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c21">X</font> = <font color="Maroon" title="c20">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c21">X</font>,<font color="Maroon" title="c12">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">L9</font></span><br/><div><span class="lab"><font color="Green" title="E44">A73</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_16_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c12">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">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="c18">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font> )</span><br/><a NAME="E1:28_16_1"/>
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a></span>;<br/><a NAME="E2:28_16_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:28_16_1"/><span class="lab"><font color="Green" title="E45">A74</font></span>: 
<font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b9</font>
 <span class="kw">and </span><br/><a NAME="E5:28_16_1"/><span class="lab"><font color="Green" title="E46">A75</font></span>: 
 <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">b9</font>
 <span class="kw">and </span><br/><a NAME="E6:28_16_1"/><span class="lab"><font color="Green" title="E47">A76</font></span>: 
<font color="Maroon" title="c23">b9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><span class="kw">assume </span><a NAME="E7:28_16_1"/><span class="lab"><font color="Green" title="E48">A77</font></span>: 
<font color="Maroon" title="c21">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font></span><br/><a NAME="E8:28_16_1"/><span class="lab"><font color="Green" title="E49">A78</font></span>: 
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">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_16_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_16_1_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">b9</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_16_1"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E2:28_16_1_1"/><span class="kw">then </span>
<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">b9</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" title="XREAL_1:th.72">XREAL_1:72</a></span>;<br/>
<a NAME="E3:28_16_1_1"/><span class="kw">then </span>
<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">b9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_16_1"><span class="lab"><font color="Green" title="E46">A75</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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">b9</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="c12">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:28_16_1_1"/><span class="lab"><font color="Green" title="E50">A79</font></span>: 
<font color="Maroon" title="c23">b9</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:28_16_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">b9</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b9</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">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E7:28_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A80</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">b9</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b9</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">b9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E8:28_16_1_1"/>
<font color="Maroon" title="c23">b9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_16_1_1"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E9:28_16_1_1"/><span class="kw">then </span>
<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">b9</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<a NAME="E10:28_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A81</font></span>: 
<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">b9</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="c21">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_16_1"><span class="lab"><font color="Green" title="E46">A75</font></span></a></span>;<br/>
<a NAME="E11:28_16_1_1"/>
<font color="Maroon" title="c23">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_16_1"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="txt" href="#E5:28_16_1_1"><span class="lab"><font color="Green" title="E50">A79</font></span></a></span>;<br/>
<a NAME="E12:28_16_1_1"/><span class="kw">then </span>
<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="c21">X</font>,<font color="Maroon" title="c12">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_16_1"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="txt" href="#E7:28_16_1"><span class="lab"><font color="Green" title="E48">A77</font></span></a>, <a class="ref" href="lattice3.html#T40" title="LATTICE3:th.40">LATTICE3:40</a></span>;<br/>
<a NAME="E13:28_16_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28_16_1_1"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></span>;<br/>
<span class="kw">then </span><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">b9</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>
<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="c4">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">c</font>,<font color="Maroon" title="c22">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="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">b9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<font color="Maroon" title="c23">b9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_16_1"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:28_16_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_16_1_1"><span class="lab"><font color="Green" title="E51">A80</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><font color="Maroon" title="c18">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="c4">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c18">z</font>,<font color="Maroon" title="c22">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">z</font>,<font color="Maroon" title="c22">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c18">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">b9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_16_1"><span class="lab"><font color="Green" title="E45">A74</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c18">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_16_1"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:28_16_1"/>
<font color="Maroon" title="c18">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c22">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_16_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c12">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c21">X</font> implies <font color="Maroon" title="c22">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c18">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_16_2"/>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c18">z</font></span><br/><a NAME="E2:28_16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A82</font></span>: 
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">y</font> &amp; <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="E3:28_16_2"/>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a></span>;<br/><a NAME="E4:28_16_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">q9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:28_16_2"/><span class="lab"><font color="Green" title="E46">A83</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q9</font>
 <span class="kw">and </span><br/><a NAME="E7:28_16_2"/>
 <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">q9</font>
 <span class="kw">and </span><br/><a NAME="E8:28_16_2"/>
<font color="Maroon" title="c23">q9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><font color="Maroon" title="c22">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c18">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="c4">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="c18">z</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c18">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c18">z</font>)

<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c23">q9</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_16_2"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="ref" href="real_lat.html#D1" title="REAL_LAT:def.1">REAL_LAT:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c22">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_16_2"><span class="lab"><font color="Green" title="E46">A83</font></span></a>, <a class="txt" href="#E2:28_16_2"><span class="lab"><font color="Green" title="E45">A82</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:28_16_2"/>
<font color="Maroon" title="c22">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c18">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:28_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c18">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></span>;<br/><a NAME="E10:28_16"/><span class="kw">then </span>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c21">X</font>,<font color="Maroon" title="c12">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_16"><span class="lab"><font color="Green" title="E44">A73</font></span></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></span>;<br/><span class="kw">hence </span><a NAME="E11:28_16"/>
 not  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> (<font color="Maroon" title="c21">X</font>,<font color="Maroon" title="c12">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">L9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:28_16"><span class="lab"><font color="Green" title="E43">A72</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E38:28"/>
 not <font color="Maroon" title="c17">L9</font> is  <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#D3" target="_self" title="MSUALG_7:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
