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

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>set </b><font color="Maroon">L'</font> =  <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/>
<b>reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">r1</font>, <font color="Maroon">R2</font> = <font color="Maroon">r2</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green">E36</font></i>: 
( 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> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></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"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></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"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><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 <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:26_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_1_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">r2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<a NAME="E3:26_1_1_1_1"/>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:26_1_1_1_1"/>
<font color="Maroon">a</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/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>;<br/>

<a NAME="E5:26_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:26_1_1_1_1"/><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> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">b1</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="E8:26_1_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E10:26_1_1_1_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 ;<br/>

<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</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">b</font>,<font color="Maroon">a</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">b</font>,<font color="Maroon">a</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">b'</font>,<font color="Maroon">r2</font>
<b>by </b><i><a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">b</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="E12:26_1_1_1_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>thus </b><a NAME="E4:26_1_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">A1</font> =  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font>;<br/>
<a NAME="E1:26_1_1_2_1"/><i><font color="Green">E65</font></i>: 
(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font> &amp; ( for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<a NAME="E2:26_1_1_2_1"/>
 ex <font color="Olive">LB</font> being   <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font> st <font color="Olive">LB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">LB</font> = <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 1 as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:26_1_1_2_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><a NAME="E2:26_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</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> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>
 ;<br/><a NAME="E3:26_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">w</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="E5:26_1_1_2_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><a NAME="E6:26_1_1_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:26_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:26_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E9:26_1_1_2_1_1_1"/>
<font color="Maroon">LB</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">LB</font> = <font color="Maroon">LB</font> as    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font> <b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">LB</font>
;<br/>

<b>thus </b><a NAME="E4:26_1_1_2_1_1"/>
<font color="Maroon">LB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_1_1_2_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">X1</font> is <a href="supinf_1.html#V4">bounded_below</a>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E4:26_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xxreal_0.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_2_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:26_1_1_2_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xxreal_0.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:26_1_1_2_1_2"/><b>then </b>
 <a href="xxreal_0.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:26_1_1_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T84">SUPINF_1:84</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:26_1_1_2_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><a NAME="E2:26_1_1_2_1_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:26_1_1_2_1_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">w</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="E5:26_1_1_2_1_3"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/><b>thus </b><a NAME="E6:26_1_1_2_1_3"/>
<font color="Maroon">R1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:26_1_1_2_1"/><b>then </b>
<font color="Maroon">R1</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E8:26_1_1_2_1"/><b>then </b>
<font color="Maroon">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">R1'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">A1'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:26_1_1_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">R1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">A1'</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">R1'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A1'</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font>;<br/>
<a NAME="E12:26_1_1_2_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:26_1_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</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="E15:26_1_1_2_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/>
<a NAME="E16:26_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:26_1_1_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">g</font> is <a href="xxreal_0.html#V1">ext-real</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E18:26_1_1_2_1"/>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E19:26_1_1_2_1"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">A'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">R2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E21:26_1_1_2_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">R2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> &amp; <font color="Maroon">A'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E22:26_1_1_2_1"/>
<font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E24:26_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_4"><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 <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>;<br/><a NAME="E1:26_1_1_2_1_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">q'</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/><a NAME="E4:26_1_1_2_1_4"/><i><font color="Green">E116</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/><b>assume </b><a NAME="E5:26_1_1_2_1_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E6:26_1_1_2_1_4"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">q1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:26_1_1_2_1_4"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> )
 <b>by </b><i/>;<br/><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<a href="real_lat.html#K1">minreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">q</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">q</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">a'</font>,<font color="Maroon">q'</font>
<b>by </b><i><a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">a</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="E10:26_1_1_2_1_4"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:26_1_1_2_1"/>
<font color="Maroon">a</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/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span>;<br/>

<a NAME="E28:26_1_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">b'</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>assume </b><a NAME="E31:26_1_1_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:26_1_1_2_1_5"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> ;<br/><a NAME="E3:26_1_1_2_1_5"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E5:26_1_1_2_1_5"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">h1</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">b'</font>,<font color="Maroon">h'</font> = 
<a href="real_lat.html#K1">minreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">h1</font>
<b>by </b><i><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> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span></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">b</font>,<font color="Maroon">h1</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/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">h1</font>

<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">h1</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">b'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><a NAME="E7:26_1_1_2_1_5"/><b>then </b>
<font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h'</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="E8:26_1_1_2_1_5"/>
<font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E33:26_1_1_2_1"/><b>then </b>
<font color="Maroon">B</font> is    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E34:26_1_1_2_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E36:26_1_1_2_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="real_lat.html#K1">minreal</a>  <a href="realset1.html#K1">||</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</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">b</font>,<font color="Maroon">a</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">b</font>,<font color="Maroon">a</font>

<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="real_lat.html#D1">REAL_LAT:def 1</a></i>
<br/>.= 
<font color="Maroon">b</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="E38:26_1_1_2_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:26"/>
 <a href="msualg_7.html#K1" target="_self">RealSubLatt</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> is <a href="lattice3.html#V4">complete</a>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D6">VECTSP_8:def 6</a></i>;<br/>


</div>
