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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>









<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> )
 ;<br/>

<a NAME="E2:55"/>
for <font color="Olive" title="b1">r1</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r1</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1_1"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c6">r1</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="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="c7">r0</font> = <font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</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/><a NAME="E4:55_1_1_1_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">r0</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1_1_1_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1_1_1_1"/>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c6">r1</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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:55_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></i>;<br/>
<a NAME="E8:55_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="extreal1.html#T44" title="EXTREAL1:th.44">EXTREAL1:44</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E9:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:55_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E6:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="mesfunc1.html#E9:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82" title="XCMPLX_1:th.82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 1
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font>

;<br/>
<a NAME="E12:55_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E13:55_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="mesfunc1.html#E8:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="mesfunc1.html#E11:55_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:55_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E46">A13</font></i>: 
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A15</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:55_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E50">A17</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E4:55_1_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></i>;<br/>
<a NAME="E7:55_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="extreal1.html#T51" title="EXTREAL1:th.51">EXTREAL1:51</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E8:55_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">r0</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A19</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font>

;<br/>
<a NAME="E10:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E2:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="mesfunc1.html#T6" target="_self" title="MESFUNC1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E11:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="mesfunc1.html#E7:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="mesfunc1.html#E9:55_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1_1_1_1_1_1_1_1"/>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E2:55_1_1_1_1_1_1_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_1_1_1_1_2"/><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1_1_1_2"/>
 <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_1_1_1_1_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E46">A21</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">y</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:55_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E48">A23</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp;  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E2:55_1_1_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="extreal1.html#T39" title="EXTREAL1:th.39">EXTREAL1:39</a></i>;<br/>
<a NAME="E7:55_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E49">A24</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_2"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="extreal1.html#T45" title="EXTREAL1:th.45">EXTREAL1:45</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E8:55_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E50">A25</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:55_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E2:55_1_1_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E51">A26</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="mesfunc1.html#E8:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E50">A25</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82" title="XCMPLX_1:th.82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 1
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font>

;<br/>
<a NAME="E11:55_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E12:55_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="txt" href="mesfunc1.html#E7:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E49">A24</font></i></a>, <a class="txt" href="mesfunc1.html#E10:55_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:55_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E46">A27</font></i>: 
 <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_2_1_2"/><i><font color="Green" title="E47">A28</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E48">A29</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:55_1_1_1_1_1_1_2_1_2"/><i><font color="Green" title="E50">A31</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E4:55_1_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E50">A31</font></i></a>, <a class="ref" href="extreal1.html#T38" title="EXTREAL1:th.38">EXTREAL1:38</a></i>;<br/>
<a NAME="E7:55_1_1_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E51">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1_1_1_2"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="extreal1.html#T52" title="EXTREAL1:th.52">EXTREAL1:52</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E8:55_1_1_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">r0</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A33</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c7">r0</font>

;<br/>
<a NAME="E10:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="mesfunc1.html#E2:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E50">A31</font></i></a>, <a class="ref" href="mesfunc1.html#T6" target="_self" title="MESFUNC1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E11:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E49">A30</font></i></a>, <a class="txt" href="mesfunc1.html#E7:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="mesfunc1.html#E9:55_1_1_1_1_1_1_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="mesfunc1.html#D14" target="_self" title="MESFUNC1:def.14">Def14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_1_1_1_1_1_1_2_1"/><b>then </b>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1_1_1_1_1_1_2_1"/>
 <a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E2:55_1_1_1_1_1_1_2_1"><i><font color="Green" title="E46">A27</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="mesfunc1.html#T33" target="_self" title="MESFUNC1:th.33">Th33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:55_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_1_2"/><i><font color="Green" title="E42">A34</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1_2_1_1"/>
( <font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1_2_1_1_1"/><i><font color="Green" title="E43">A35</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_1_2_1_1_1_1"/><i><font color="Green" title="E44">A36</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:55_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_1_2_1_1_1_1_1"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E45">A37</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:55_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="mesfunc1.html#E1:55_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E45">A37</font></i></a></i>;<br/>
<a NAME="E4:55_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E46">A38</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x1</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/>
<font color="Maroon" title="c9">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc1.html#E4:55_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>
;<br/>
<a NAME="E7:55_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2_1_1_1"><i><font color="Green" title="E43">A35</font></i></a>, <a class="txt" href="mesfunc1.html#E4:55_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:55_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E45">A37</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2_1_1_1_1"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D11" target="_self" title="MESFUNC1:def.11">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_1_2_1_1_2"/><i><font color="Green" title="E43">A39</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_2_1_1_2"/>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_1_2_1_1_2_1"/>
 <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:55_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E44">A40</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E4:55_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E45">A41</font></i>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  ex <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> ) )
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E44">A40</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E6:55_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E46">A42</font></i>: 
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E44">A40</font></i></a>, <a class="ref" href="mesfunc1.html#D12" target="_self" title="MESFUNC1:def.12">Def12</a></i>;<br/>
<a NAME="E7:55_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E47">A43</font></i>: 
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E4:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E45">A41</font></i></a>, <a class="txt" href="mesfunc1.html#E6:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E46">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:55_1_1_1_2_1_1_2_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/>not <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:55_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E48">A44</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<font color="Maroon" title="c9">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc1.html#E3:55_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="mesfunc1.html#D6" target="_self" title="MESFUNC1:def.6">Def6</a></i>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E5:55_1_1_1_2_1_1_2_1_1"/>
not <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font>
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:55_1_1_1_2_1_1_2"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:55_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc1.html#E6:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="mesfunc1.html#E7:55_1_1_1_2_1_1_2_1"><i><font color="Green" title="E47">A43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:55_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></i>;<br/><b>hence </b><a NAME="E4:55_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D11" target="_self" title="MESFUNC1:def.11">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<font color="Maroon" title="c5">r</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></i>;<br/>


</div>
