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<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#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</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>assume </b><b>that </b><br/><a NAME="E1:48"/><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>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E42">A2</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="c5">B</font>
 ;<br/>

<a NAME="E3:48"/>
for <font color="Olive" title="b1">r</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  <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</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="48_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:48_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#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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#E1:48"><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/>
<a NAME="E2:48_1"/><b>then </b><i><font color="Green" title="E43">A3</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#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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="mesfunc1.html#D11" target="_self" title="MESFUNC1:def.11">Def11</a></i>;<br/>
<a NAME="E3:48_1"/>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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#E2:48"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></i>;<br/>
<a NAME="E4:48_1"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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="mesfunc1.html#D11" target="_self" title="MESFUNC1:def.11">Def11</a></i>;<br/>
<a NAME="E5:48_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</font></span>)</span></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="txt" href="mesfunc1.html#E2:48_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="finsub_1.html#D1" title="FINSUB_1:def.1">FINSUB_1:def 1</a></i>;<br/>
<a NAME="E6:48_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K14" title="MESFUNC1:func.14">less_dom</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">r</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>
<b>hence </b><a NAME="E4:48"/>
<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> <a href="measure1.html#K2" title="MEASURE1:func.2">\/</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17" target="_self" title="MESFUNC1:def.17">Def17</a></i>;<br/>


</div>
