<?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#M1">Element</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:29"/>
for <font color="Olive">r</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">less_dom</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>reconsider </b><font color="Maroon">r</font> = <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><a NAME="E1:29_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></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"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</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">r</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E4:29_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E5:29_1_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <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="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E6:29_1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="supinf_2.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">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>
<a NAME="E7:29_1_1_1_1"/>
 <a href="supinf_2.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>  <a href="real_1.html#K1">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T3">SUPINF_2:3</a></i>;<br/>
<a NAME="E8:29_1_1_1_1"/><b>then </b>
(  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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/>;<br/>
<a NAME="E9:29_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">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">r</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a>, <a class="ref" href="mesfunc1.html#D14">MESFUNC1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></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>
<a NAME="E2:29_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:29_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">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">r</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:29_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">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">.</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">r</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></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="E5:29_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</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">r</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</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="E7:29_1_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp;  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D14">MESFUNC1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E9:29_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E10:29_1_1_1_2"/>
 <a href="supinf_2.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>  <a href="real_1.html#K1">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T3">SUPINF_2:3</a></i>;<br/>
<a NAME="E11:29_1_1_1_2"/><b>then </b>
 <a href="supinf_2.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>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>
 
;<br/>
<a NAME="E12:29_1_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <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="extreal2.html#T59">EXTREAL2:59</a></i>;<br/>
<a NAME="E13:29_1_1_1_2"/><b>then </b>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:29_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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#D12">MESFUNC1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1_1_1"/>
 <a href="mesfunc1.html#K15">less_dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:29_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<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="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc1.html#T36">MESFUNC1:36</a></i>;<br/><b>hence </b><a NAME="E3:29_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">less_dom</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>,<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="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/>


</div>
