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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:55"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc1.html#R2" target="_self">is_measurable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:55"/>
for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">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">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">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">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#R1">=</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">E27</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E3:55_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">r0</font> = <font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E5:55_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="xcmplx_1.html#T88">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">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</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"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E3:55_1_1_1_1_1_1_1"/>
 <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">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">E51</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">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">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, , <a class="ref" href="extreal1.html#T39">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">E53</font></i>: 
<font color="Maroon">y</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T44">EXTREAL1:44</a></i>;<br/>
<a NAME="E9:55_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">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">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
<b>by </b><i><a class="txt" href="mesfunc1.html#E6:55_1_1_1_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> 1
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r1</font>

;<br/>
<a NAME="E12:55_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E5:55_1_1_1_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E13:55_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E14:55_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc1.html#E8:55_1_1_1_1_1_1_1_1_1"><i><font color="Green">E53</font></i></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">E56</font></i>: 
 <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">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">E67</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="extreal1.html#T38">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">E114</font></i>: 
<font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T51">EXTREAL1:51</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">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>
<br/>.= 
<font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 1
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r0</font>

;<br/>
<a NAME="E10:55_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, , , </i>;<br/>
<b>hence </b><a NAME="E12:55_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 <b>by </b><i>, , </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" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">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" target="_self">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:55_1_1_1_1_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T2" target="_self">Th2</a>, <a class="txt" href="mesfunc1.html#E2:55_1_1_1_1_1_1_1"><i><font color="Green">E38</font></i></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"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E3:55_1_1_1_1_1_1_2"/>
 <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_1_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> ) )
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">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">E119</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a>, <a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green">E24</font></i></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">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>, , <a class="ref" href="extreal1.html#T39">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">E120</font></i>: 
<font color="Maroon">y</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T45">EXTREAL1:45</a></i>;<br/>
<a NAME="E8:55_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">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">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="xcmplx_0.html#K7">/</a> 1
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r1</font>

;<br/>
<a NAME="E11:55_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:55_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:55_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i>, , </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">E123</font></i>: 
 <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1_1_1_2_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">E125</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">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">E127</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="extreal1.html#T38">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">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T52">EXTREAL1:52</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">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span></span>)</span> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r0</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>
<br/>.= 
<font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 1
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r0</font>

;<br/>
<a NAME="E10:55_1_1_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="extreal1.html#K3">/</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, , , </i>;<br/>
<b>hence </b><a NAME="E12:55_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="mesfunc1.html#E1:55"><i><font color="Green">E24</font></i></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" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">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" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:55_1_1_1_1_1_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:55_1_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">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">E130</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</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">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&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">E131</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E132</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:55_1_1_1_2_1_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:55_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E4:55_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x1</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E6:55_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E135</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<i><font color="Green">E136</font></i>: <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>
;<br/>
<a NAME="E8:55_1_1_1_2_1_1_1_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, , <a class="ref" href="mesfunc1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:55_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:55_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">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">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">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">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D10">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">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D1" target="_self">Def1</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">E137</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_1_2_1_1_2"/>
 <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:55_1_1_1_2_1_1_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E4:55_1_1_1_2_1_1_2_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x1</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E6:55_1_1_1_2_1_1_2_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E7:55_1_1_1_2_1_1_2_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:55_1_1_1_2_1_1_2_1"/>
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> holds <br/>not <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">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">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:55_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:55_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E143</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:55_1_1_1_2_1_1_2_1_1"/>
not <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T3" target="_self">Th3</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="ref" href="mesfunc1.html#T4" target="_self">Th4</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">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><b>hence </b><a NAME="E4:55_1_1_1_2_1_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D1" target="_self">Def1</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">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">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">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" target="_self">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<font color="Maroon">r</font> <a href="mesfunc1.html#K8" target="_self">(#)</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2" target="_self">is_measurable_on</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div>
