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

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">x</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1"/>
( <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" title="MEASURE5:def.9">MEASURE5:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>suppose </b></a><a NAME="E1:66_1_1"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>
 ;<br/><div class="add"><a NAME="E2:66_1_1"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T62" target="_self" title="MEASURE6:th.62">Th62</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:66_1_1"/><i><font color="Green" title="E49">A3</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="c6">b</font> &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"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="measure5.html#D5" title="MEASURE5:def.5">MEASURE5:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><a NAME="E6:66_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <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"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1_1_1"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:66_1_1_1_1"/><i><font color="Green" title="E52">A6</font></i>: 
( <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">d1</font> = <font color="Maroon" title="c11">d</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/>
<a NAME="E6:66_1_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_1_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></i>;<br/>
<a NAME="E7:66_1_1_1_1"/><b>then </b><i><font color="Green" title="E53">A7</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">t</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E8:66_1_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_1_1_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_1_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:66_1_1_1_1"><i><font color="Green" title="E53">A7</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>

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:66_1_1_1"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c1</font> = <font color="Maroon" title="c10">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/>
<a NAME="E5:66_1_1_1"/>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:66_1_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></i>;<br/>
<a NAME="E6:66_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E7:66_1_1_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/>
<a NAME="E8:66_1_1_1"/>
<font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:66_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="measure6.html#E7:66_1_1_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="measure5.html#D2" title="MEASURE5:def.2">MEASURE5:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2_1"><b>suppose </b></a><a NAME="E1:66_1_1_2_1"/><i><font color="Green" title="E51">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/><a NAME="E3:66_1_1_2_1"/><b>then </b><i><font color="Green" title="E52">A11</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="measure6.html#E6:66_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">MEASURE5:53</a></i>;<br/><a NAME="E4:66_1_1_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1_2_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="measure6.html#T10" target="_self" title="MEASURE6:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E5:66_1_1_2_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="measure6.html#E1:66_1_1_2_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="measure6.html#E3:66_1_1_2_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">MEASURE5:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2_2"><b>suppose </b></a><a NAME="E1:66_1_1_2_2"/><i><font color="Green" title="E51">A12</font></i>: 
<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>
 ;<br/><div class="add"><a NAME="E2:66_1_1_2_2"/><b>then </b><i><font color="Green" title="E52">A13</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">MEASURE5:53</a></i>;<br/><a NAME="E3:66_1_1_2_2"/>
<font color="Maroon" title="c8">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_1_2_2"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:66_1_1_2_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="measure6.html#E6:66_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="measure6.html#E2:66_1_1_2_2"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="measure5.html#T53" title="MEASURE5:th.53">MEASURE5:53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><b>suppose </b></a><a NAME="E1:66_1_2"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 ;<br/><div class="add"><a NAME="E2:66_1_2"/><b>then </b><i><font color="Green" title="E48">A15</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T63" target="_self" title="MEASURE6:th.63">Th63</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:66_1_2"/><i><font color="Green" title="E49">A16</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="c6">b</font> &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#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="measure5.html#D6" title="MEASURE5:def.6">MEASURE5:def 6</a></i>;<br/><b>set </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>;<br/><b>set </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>;<br/><a NAME="E5:66_1_2"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <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"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_2_1_1"/><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:66_1_2_1_1"/><i><font color="Green" title="E52">A19</font></i>: 
( <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2_1_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">d1</font> = <font color="Maroon" title="c11">d</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/>
<a NAME="E6:66_1_2_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c12">d1</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="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_2_1_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E7:66_1_2_1_1"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/>
<a NAME="E8:66_1_2_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_2_1_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_2_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:66_1_2_1_1"><i><font color="Green" title="E53">A20</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>

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:66_1_2_1"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c1</font> = <font color="Maroon" title="c10">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/>
<a NAME="E5:66_1_2_1"/>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c11">c1</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:66_1_2_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E6:66_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/>
<a NAME="E7:66_1_2_1"/><b>then </b><i><font color="Green" title="E52">A22</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="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</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="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/>
<a NAME="E8:66_1_2_1"/>
<font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:66_1_2_1"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="measure6.html#E7:66_1_2_1"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1_2_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_2_2"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_2_1"><b>suppose </b></a><a NAME="E1:66_1_2_2_1"/><i><font color="Green" title="E51">A23</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:66_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/><a NAME="E3:66_1_2_2_1"/><b>then </b><i><font color="Green" title="E52">A24</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">MEASURE5:54</a></i>;<br/><a NAME="E4:66_1_2_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2_2_1"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="measure6.html#T10" target="_self" title="MEASURE6:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E5:66_1_2_2_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="measure6.html#E1:66_1_2_2_1"><i><font color="Green" title="E51">A23</font></i></a>, <a class="txt" href="measure6.html#E3:66_1_2_2_1"><i><font color="Green" title="E52">A24</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">MEASURE5:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_2_2"><b>suppose </b></a><a NAME="E1:66_1_2_2_2"/><i><font color="Green" title="E51">A25</font></i>: 
<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>
 ;<br/><div class="add"><a NAME="E2:66_1_2_2_2"/><b>then </b><i><font color="Green" title="E52">A26</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">MEASURE5:54</a></i>;<br/><a NAME="E3:66_1_2_2_2"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_2_2_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:66_1_2_2_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="measure6.html#E2:66_1_2_2_2"><i><font color="Green" title="E52">A26</font></i></a>, <a class="ref" href="measure5.html#T54" title="MEASURE5:th.54">MEASURE5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3"><b>suppose </b></a><a NAME="E1:66_1_3"/><i><font color="Green" title="E47">A27</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:66_1_3"/><b>then </b><i><font color="Green" title="E48">A28</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T64" target="_self" title="MEASURE6:th.64">Th64</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:66_1_3"/><i><font color="Green" title="E49">A29</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="c6">b</font> &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#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="measure5.html#D7" title="MEASURE5:def.7">MEASURE5:def 7</a></i>;<br/><b>set </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>;<br/><b>set </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>;<br/><a NAME="E5:66_1_3"/><i><font color="Green" title="E50">A30</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <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"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_3_1"/>
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_3_1_1"/><i><font color="Green" title="E51">A31</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:66_1_3_1_1"/><i><font color="Green" title="E52">A32</font></i>: 
( <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3_1_1"><i><font color="Green" title="E51">A31</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">d1</font> = <font color="Maroon" title="c11">d</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/>
<a NAME="E6:66_1_3_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_3"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_3_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/>
<a NAME="E7:66_1_3_1_1"/><b>then </b><i><font color="Green" title="E53">A33</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a>, <a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E8:66_1_3_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_3_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_3_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:66_1_3_1_1"><i><font color="Green" title="E53">A33</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>

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:66_1_3_1"/><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c1</font> = <font color="Maroon" title="c10">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/>
<a NAME="E5:66_1_3_1"/>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c11">c1</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:66_1_3_1"><i><font color="Green" title="E51">A34</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/>
<a NAME="E6:66_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a>, <a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E7:66_1_3_1"/><b>then </b><i><font color="Green" title="E52">A35</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="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/>
<a NAME="E8:66_1_3_1"/>
<font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:66_1_3_1"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_3"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="measure6.html#E7:66_1_3_1"><i><font color="Green" title="E52">A35</font></i></a>, <a class="ref" href="measure5.html#D4" title="MEASURE5:def.4">MEASURE5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1_3_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_3_2"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_2_1"><b>suppose </b></a><a NAME="E1:66_1_3_2_1"/><i><font color="Green" title="E51">A36</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:66_1_3_2_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/><a NAME="E3:66_1_3_2_1"/><b>then </b><i><font color="Green" title="E52">A37</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_3"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_3"><i><font color="Green" title="E50">A30</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">MEASURE5:55</a></i>;<br/><a NAME="E4:66_1_3_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3_2_1"><i><font color="Green" title="E51">A36</font></i></a>, <a class="ref" href="measure6.html#T10" target="_self" title="MEASURE6:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E5:66_1_3_2_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_3"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="measure6.html#E1:66_1_3_2_1"><i><font color="Green" title="E51">A36</font></i></a>, <a class="txt" href="measure6.html#E3:66_1_3_2_1"><i><font color="Green" title="E52">A37</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">MEASURE5:55</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_3_2_2"><b>suppose </b></a><a NAME="E1:66_1_3_2_2"/><i><font color="Green" title="E51">A38</font></i>: 
<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>
 ;<br/><div class="add"><a NAME="E2:66_1_3_2_2"/><b>then </b><i><font color="Green" title="E52">A39</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_3"><i><font color="Green" title="E49">A29</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">MEASURE5:55</a></i>;<br/><a NAME="E3:66_1_3_2_2"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_3_2_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:66_1_3_2_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_3"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_3"><i><font color="Green" title="E50">A30</font></i></a>, <a class="txt" href="measure6.html#E2:66_1_3_2_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="measure5.html#T55" title="MEASURE5:th.55">MEASURE5:55</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4"><b>suppose </b></a><a NAME="E1:66_1_4"/><i><font color="Green" title="E47">A40</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>
 ;<br/><div class="add"><a NAME="E2:66_1_4"/><b>then </b><i><font color="Green" title="E48">A41</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a>
 <b>by </b><i><a class="ref" href="measure6.html#T65" target="_self" title="MEASURE6:th.65">Th65</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E4:66_1_4"/><i><font color="Green" title="E49">A42</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="c6">b</font> &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#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="measure5.html#D8" title="MEASURE5:def.8">MEASURE5:def 8</a></i>;<br/><b>set </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>;<br/><b>set </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>;<br/><a NAME="E5:66_1_4"/><i><font color="Green" title="E50">A43</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <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"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1_4_1"/>
<font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_4_1_1"/><i><font color="Green" title="E51">A44</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:66_1_4_1_1"/><i><font color="Green" title="E52">A45</font></i>: 
( <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4_1_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">d1</font> = <font color="Maroon" title="c11">d</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/>
<a NAME="E6:66_1_4_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c12">d1</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="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_4_1_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="measure5.html#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></i>;<br/>
<a NAME="E7:66_1_4_1_1"/><b>then </b><i><font color="Green" title="E53">A46</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a>, <a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E8:66_1_4_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_4_1_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_4_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 <b>by </b><i><a class="txt" href="measure6.html#E7:66_1_4_1_1"><i><font color="Green" title="E53">A46</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>

<b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:66_1_4_1"/><i><font color="Green" title="E51">A47</font></i>: 
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c9">c</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c1</font> = <font color="Maroon" title="c10">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/>
<a NAME="E5:66_1_4_1"/>
( <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:66_1_4_1"><i><font color="Green" title="E51">A47</font></i></a>, <a class="ref" href="measure5.html#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></i>;<br/>
<a NAME="E6:66_1_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a>, <a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/>
<a NAME="E7:66_1_4_1"/><b>then </b><i><font color="Green" title="E52">A48</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</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="ref" href="measure6.html#T8" target="_self" title="MEASURE6:th.8">Th8</a></i>;<br/>
<a NAME="E8:66_1_4_1"/>
<font color="Maroon" title="c11">c1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5" title="SUPINF_2:th.5">SUPINF_2:5</a></i>;<br/>
<a NAME="E9:66_1_4_1"/><b>then </b>
( <font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="measure6.html#E4:66_1_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="measure6.html#E7:66_1_4_1"><i><font color="Green" title="E52">A48</font></i></a>, <a class="ref" href="measure5.html#D3" title="MEASURE5:def.3">MEASURE5:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1_4_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_4_2"/>
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_1"><b>suppose </b></a><a NAME="E1:66_1_4_2_1"/><i><font color="Green" title="E51">A49</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:66_1_4_2_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="measure6.html#T12" target="_self" title="MEASURE6:th.12">Th12</a></i>;<br/><a NAME="E3:66_1_4_2_1"/><b>then </b><i><font color="Green" title="E52">A50</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_4"><i><font color="Green" title="E48">A41</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">MEASURE5:56</a></i>;<br/><a NAME="E4:66_1_4_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="measure6.html#T10" target="_self" title="MEASURE6:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E5:66_1_4_2_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="measure6.html#E1:66_1_4_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="measure6.html#E3:66_1_4_2_1"><i><font color="Green" title="E52">A50</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">MEASURE5:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_4_2_2"><b>suppose </b></a><a NAME="E1:66_1_4_2_2"/><i><font color="Green" title="E51">A51</font></i>: 
<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>
 ;<br/><div class="add"><a NAME="E2:66_1_4_2_2"/><b>then </b><i><font color="Green" title="E52">A52</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="measure6.html#E4:66_1_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">MEASURE5:56</a></i>;<br/><a NAME="E3:66_1_4_2_2"/>
<font color="Maroon" title="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</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="c4">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:66_1_4_2_2"><i><font color="Green" title="E51">A51</font></i></a>, <a class="ref" href="measure6.html#T11" target="_self" title="MEASURE6:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E4:66_1_4_2_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure6.html#E2:66_1_4"><i><font color="Green" title="E48">A41</font></i></a>, <a class="txt" href="measure6.html#E5:66_1_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="measure6.html#E2:66_1_4_2_2"><i><font color="Green" title="E52">A52</font></i></a>, <a class="ref" href="measure5.html#T56" title="MEASURE5:th.56">MEASURE5:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
