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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="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>











<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> )
 ;<br/>

<a NAME="E2:75"/><b>then </b>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="mesfunc6.html#D6" target="_self" title="MESFUNC6:def.6">Def6</a></i>;<br/>
<a NAME="E3:75"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">less_dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c6">s</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#D17" title="MESFUNC1:def.17">MESFUNC1:def 17</a></i>;<br/>
<a NAME="E4:75"/><b>then </b><i><font color="Green" title="E28">A2</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="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font></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" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E5:75"/>
<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="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></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="mesfunc6.html#E1:75"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="mesfunc6.html#T13" target="_self" title="MESFUNC6:th.13">Th13</a></i>;<br/>
<a NAME="E6:75"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></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" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E7:75"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
<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="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <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="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</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="txt" href="mesfunc6.html#E4:75"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>;<br/>
<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="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <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="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="measure1.html#K3" title="MEASURE1:func.3">/\</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E9:75"/>
<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="mesfunc6.html#K4" title="MESFUNC6:func.4">great_eq_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">less_dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">s</font></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="mesfunc6.html#E7:75"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="mesfunc1.html#D11" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/>


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