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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> &amp; ( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>suppose </b></a><a NAME="E1:55_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2"><b>suppose </b></a><a NAME="E1:55_1_2"/><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> ;<br/><a NAME="E2:55_1_2"/>
not <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_2_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_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="E1:55_1_2_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span><b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:55_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T52" target="_self" title="MEASURE6:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T36" target="_self" title="MEASURE6:th.36">Th36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:55_1_2_1_1_2"/><b>then </b>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; ( <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a> </span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T52" target="_self" title="MEASURE6:th.52">Th52</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T37" target="_self" title="MEASURE6:th.37">Th37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D4" title="XXREAL_0:def.4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:55_1_2"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> holds <br/>for <font color="Olive" title="b3">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:55_1_2_2"/><i><font color="Green" title="E42">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E2:55_1_2_2"/><i><font color="Green" title="E43">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_2_2_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) or ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) )
 </span><b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:55_1_2_2_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="measure6.html#T46" target="_self" title="MEASURE6:th.46">Th46</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:55_1_2_2_1_2"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="measure6.html#T55" target="_self" title="MEASURE6:th.55">Th55</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:55_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:55_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:55_1_2_2_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:55_1_2_2_1_3"/><i><font color="Green" title="E44">A7</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_2_2_1_3_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 </span><b>by </b><i><a class="txt" href="measure6.html#E1:55"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_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="E1:55_1_2_2_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2_2_1_3_1_1_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="measure6.html#T46" target="_self" title="MEASURE6:th.46">Th46</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E46">A9</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">c</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c7">c</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c4">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E4:55_1_2_2_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/><a NAME="E5:55_1_2_2_1_3_1_1_1_1_1_2"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="measure6.html#T54" target="_self" title="MEASURE6:th.54">Th54</a></i>;<br/><a NAME="E6:55_1_2_2_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2_2_1_3_1_1_1_1_1_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="measure6.html#E4:55_1_2_2_1_3_1_1_1_1_1_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_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="E1:55_1_2_2_1_3_1_1_1_1_1_2_1_1"/>
( (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  ) or (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  ) )
 </span><b>by </b><i><a class="txt" href="measure6.html#E6:55_1_2_2_1_3_1_1_1_1_1_2"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_1_1_1_2_1_1_1"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T57" target="_self" title="MEASURE6:th.57">Th57</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_3_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E49">A12</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="measure6.html#T54" target="_self" title="MEASURE6:th.54">Th54</a></i>;<br/><a NAME="E3:55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="measure6.html#E2:55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E49">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E4:55_1_2_2_1_3_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:55_1_2_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_2_2_1_3_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_2"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_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="E1:55_1_2_2_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c7">c</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c4">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E4:55_1_2_2_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E47">A15</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/><a NAME="E5:55_1_2_2_1_3_1_1_2_1_1_1"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="measure6.html#T54" target="_self" title="MEASURE6:th.54">Th54</a></i>;<br/><a NAME="E6:55_1_2_2_1_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E48">A16</font></i>: 
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_3_1_1_2_1_1_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="measure6.html#E4:55_1_2_2_1_3_1_1_2_1_1_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_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="E1:55_1_2_2_1_3_1_1_2_1_1_1_1_1"/>
( (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  ) or (  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  ) )
 </span><b>by </b><i><a class="txt" href="measure6.html#E6:55_1_2_2_1_3_1_1_2_1_1_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_2_1_1_1_1_1_1"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="measure6.html#T57" target="_self" title="MEASURE6:th.57">Th57</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_3_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"/>
(  <a href="measure6.html#K4" title="MEASURE6:func.4">^^</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">A</font> <a href="measure6.html#K5" title="MEASURE6:func.5">^^</a>  )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="measure6.html#T54" target="_self" title="MEASURE6:th.54">Th54</a></i>;<br/><a NAME="E3:55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="measure6.html#E2:55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E4:55_1_2_2_1_3_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:55_1_2_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:55_1_2_2_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_3_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:55_1_2_2_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font>
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_3_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2_2_1_3"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="measure6.html#E1:55_1_2_2_1_3_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="measure6.html#T46" target="_self" title="MEASURE6:th.46">Th46</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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="E3:55_1_2_2_1_3_1_1_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_2_2_1_3"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:55_1_2_2_1_4"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ;<br/><div class="add"><a NAME="E2:55_1_2_2_1_4"/><b>then </b>
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E2:55_1_2_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="measure6.html#T46" target="_self" title="MEASURE6:th.46">Th46</a></i>;<br/><b>hence </b><a NAME="E3:55_1_2_2_1_4"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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="E5:55_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> is   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T46" target="_self" title="MEASURE6:th.46">Th46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
