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

<b>set </b><font color="Maroon">R</font> =  <a href="real_lat.html#K3">Real_Lattice</a> ;<br/>
<a NAME="E1:29"/><i><font color="Green">E34</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1;<br/>
<a NAME="E3:29"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1</span>)</span> &amp; the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Ma</font> = <a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">Mi</font> = <a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font>;<br/>
<b>reconsider </b><font color="Maroon">Ma</font> = <a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font>, <font color="Maroon">Mi</font> = <a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1 as  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<a NAME="E6:29"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:29_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E5:29_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:29_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:29_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:29_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:29_1_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_1_2_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E4:29_1_2_1_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E5:29_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:29_1_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>


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


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E4:29_2_1_1_2"/>
<font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:29_2_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:29_2_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:29_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:29_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:29"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ma</font> = <font color="Maroon">Ma</font> <a href="realset1.html#K1">||</a> <font color="Maroon">B</font>, <font color="Maroon">mi</font> = <font color="Maroon">Mi</font> <a href="realset1.html#K1">||</a> <font color="Maroon">B</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E12:29"/><i><font color="Green">E109</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ma</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:29_3"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1_1_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_1_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> 1,1
<b>by </b><i>, , <a class="ref" href="real_lat.html#D2">REAL_LAT:def 2</a></i>
<br/>.= 
1

;<br/><a NAME="E4:29_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:29_3_1_1_1_1_1_1"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_1_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_1_1_1_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> 1,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E5:29_3_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><a NAME="E6:29_3_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_3_1_1_1_1_1_2"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_3_1_1_1"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_3_1_1_2_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_3_1_1_2_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">y1</font>,1
<b>by </b><i>, , <a class="ref" href="real_lat.html#D2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E4:29_3_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><a NAME="E5:29_3_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  or <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_3_1_1_2_1_1_1"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_3_1_1_2_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_3_1_1_2_1_1_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Ma</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D2">REAL_LAT:def 2</a></i>
;<br/><a NAME="E5:29_3_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><a NAME="E6:29_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:29_3_1_1_2_1_1_2"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:29_3_1_1_2"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:29_3"/>
<font color="Maroon">ma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ma</font> = <font color="Maroon">ma</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_4"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">mi</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_4"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:29_4"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_4_1_1_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1_1_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_1_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 1,1
<b>by </b><i>, , <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
1

;<br/><a NAME="E4:29_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:29_4_1_1_1_1_1_1"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_1_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_4_1_1_1_1_1_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 1,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E5:29_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><a NAME="E6:29_4_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_4_1_1_1_1_1_2"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_4_1_1_1"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_4_1_1_2"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_4_1_1_2_1_1"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_4_1_1_2_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">y1</font>,1
<b>by </b><i>, , <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E4:29_4_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><a NAME="E5:29_4_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  or <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:29_4_1_1_2_1_1_1"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_4_1_1_2_1_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_4_1_1_2_1_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">Mi</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, , <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
;<br/><a NAME="E5:29_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> or <font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><a NAME="E6:29_4_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:29_4_1_1_2_1_1_2"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:29_4_1_1_2"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:29_4"/>
<font color="Maroon">mi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">mi</font> = <font color="Maroon">mi</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L'</font> =  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">B</font>,<font color="Maroon">ma</font>,<font color="Maroon">mi</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a>  ;<br/>
<div><i><font color="Green">E148</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:29_5"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:29_5"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_5"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:29_5"/>
<font color="Maroon">x1</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a> 
 <b>by </b><i>, <a class="ref" href="real_lat.html#D4">REAL_LAT:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E150</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_6_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1_1"><b>suppose </b></a><a NAME="E1:29_6_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_6_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_6_1_1"/>
<font color="Maroon">x1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_6_1_2"><b>suppose </b></a><a NAME="E1:29_6_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:29_6_1_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E4:29_6_1_2"/>
<font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:29_6_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:29_6_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:29_6_1_2"/>
<font color="Maroon">x1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><a NAME="E1:29_7"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L'</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L'</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q'</font>,<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T8">REAL_LAT:8</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font>, <font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T10">REAL_LAT:10</a></i>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">q'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="real_lat.html#T12">REAL_LAT:12</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q'</font>,<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T9">REAL_LAT:9</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E158</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font>, <font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="real_lat.html#T11">REAL_LAT:11</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">q</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">r</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="real_lat.html#K3">Real_Lattice</a>  <b>by </b><i>, </i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><a href="real_lat.html#K2">maxreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="real_lat.html#T13">REAL_LAT:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E27:29"/><b>then </b>
( <font color="Maroon">L'</font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">L'</font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">L'</font> is <a href="lattices.html#V8">meet-absorbing</a> &amp; <font color="Maroon">L'</font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">L'</font> is <a href="lattices.html#V7">meet-associative</a> &amp; <font color="Maroon">L'</font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i>, , <a class="txt" href="msualg_7.html#E1:29"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="msualg_7.html#E3:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L'</font> as   <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">K</font> =  <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1;<br/>
<b>reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L'</font> as    <a href="nat_lat.html#M2">SubLattice</a> of  <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> 0,1 <b>by </b><i>, , <a class="ref" href="nat_lat.html#D16">NAT_LAT:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">L'</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L'</font>;<br/><b>thus </b><a NAME="E1:29_14"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_14_1"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:29_14_1"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_14_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:29_14_1"/><i><font color="Green">E160</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>assume </b><a NAME="E6:29_14_1"/><i><font color="Green">E161</font></i>: 
not  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 ;<br/>

<a NAME="E7:29_14_1"/><b>then </b>
( not  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; not  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:29_14_1"/><b>then </b><i><font color="Green">E162</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_14_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:29_14_1_1"/><i><font color="Green">E164</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/> <a href="xxreal_0.html#K3">min</a> <font color="Maroon">z1</font>,<font color="Maroon">y</font> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a>, <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>

<br/>.= 
<font color="Maroon">z</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a>, , <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><a NAME="E7:29_14_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E8:29_14_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:29_14_1_1"/><b>then </b>
for <font color="Olive">v</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>   holds <br/>(  not <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">v</font> or  not <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:29_14_1_1"/><b>then </b>
not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E11:29_14_1_1"/>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:29_14_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:29_14_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_14_1_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_14_1_2_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:29_14_1_2_1_1"/><b>then </b>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">w</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font>
 ;<br/><a NAME="E3:29_14_1_2_1_1"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">w</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_14_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:29_14_1_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E2:29_14_1_2_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_14_1_2_1_1_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_14_1_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 ;<br/><font color="Maroon">r1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">r1</font>,<font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">r1</font>,<font color="Maroon">w</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">e</font>,1
<b>by </b><i>, <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">r1</font>
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><b>hence </b><a NAME="E7:29_14_1_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div><a NAME="E5:29_14_1_2_1_1"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/><a NAME="E6:29_14_1_2_1_1"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_14_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_14_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_14_1_2_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_14_1_2_1_2"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>;<br/><a NAME="E3:29_14_1_2_1_2"/><b>then </b>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_14_1_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:29_14_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E31:29"/>
<font color="Maroon">L'</font> is <a href="msualg_7.html#V1" target="_self">/\-inheriting</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> ;<br/><a NAME="E1:29_15"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L'</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>take </b><font color="Maroon">X</font> = <font color="Maroon">X</font>;<br/><a NAME="E3:29_15"/>
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon">z</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:29_15_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E2:29_15_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">q'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_15_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">q'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_15_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E7:29_15_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">z</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">q'</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a>, <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a>, , <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><b>hence </b><a NAME="E9:29_15_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div><a NAME="E6:29_15"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:29_15_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E2:29_15_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">b'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:29_15_2"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>assume </b><a NAME="E5:29_15_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E6:29_15_2"/><i><font color="Green">E174</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_15_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_15_2_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/>

<a NAME="E2:29_15_2_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:29_15_2_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E4:29_15_2_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E6:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E8:29_15_2_1"/>
<font color="Maroon">b'</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:29_15_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E10:29_15_2_1"/><b>then </b><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:29_15_2_1"/>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:29_15_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#T41">LATTICE3:41</a></i>;<br/>
<a NAME="E13:29_15_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 = 
<font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>

<br/>.= 
 <a href="xxreal_0.html#K3">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#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b'</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">b'</font>
<b>by </b><i>, <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E15:29_15_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>


</div><b>end;</b></div><font color="Maroon">z</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Maroon">b</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<font color="Maroon">b'</font>
<b>by </b><i>, <a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="msualg_7.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><b>hence </b><a NAME="E8:29_15_2"/>
<font color="Maroon">z</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div><a NAME="E8:29_15"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self">Th2</a>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/><a NAME="E9:29_15"/><i><font color="Green">E179</font></i>: 
not 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:29_15"/>
for <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>   holds <br/>(  not <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 or  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> or  not <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><a NAME="E11:29_15"/><b>then </b>
not 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E12:29_15"/>
not  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E33:29"/>
not <font color="Maroon">L'</font> is <a href="msualg_7.html#V2" target="_self">\/-inheriting</a>
 <b>by </b><i/>;<br/>


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