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

<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E40">A1</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> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E2:54"/><i><font color="Green" title="E41">A2</font></i>: 
(  <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="c2">a</font> &amp; <font color="Maroon" title="c2">a</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/>

<a NAME="E3:54"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="measure6.html#T15" target="_self" title="MEASURE6:th.15">Th15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <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> )
 </span><b>by </b><i><a class="txt" href="measure6.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="measure6.html#T52" target="_self" title="MEASURE6:th.52">Th52</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1"/>
<font color="Maroon" title="c2">a</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#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="measure6.html#E3:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_2"/>
<font color="Maroon" title="c2">a</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#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="measure6.html#E3:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="measure5.html#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3"><b>suppose </b></a><a NAME="E1:54_1_3"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_3"/>
<font color="Maroon" title="c2">a</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#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="measure6.html#E3:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_4"><b>suppose </b></a><a NAME="E1:54_1_4"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_4"/>
<font color="Maroon" title="c2">a</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#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="measure6.html#E3:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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