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<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">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>









<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>now </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/><a NAME="E1:65_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">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</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="mesfunc5.html#T9" target="_self">Th9</a>, <a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/><a NAME="E2:65_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><a NAME="E3:65_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></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="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><b>hence </b><a NAME="E4:65_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> <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>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:65_1"/>
<font color="Maroon">f</font> <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><b>end;</b></div>
<b>hence </b><a NAME="E2:65"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> implies <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 ;<br/>

<b>assume </b><a NAME="E3:65"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </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/><a NAME="E1:65_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K3">/\</a> <font color="Maroon">A</font></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="mesfunc5.html#T9" target="_self">Th9</a>, <a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/><a NAME="E2:65_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></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/><b>hence </b><a NAME="E3:65_2"/>
<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">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</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/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <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/>


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