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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>, <font color="Maroon" title="c4">g</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="c5">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>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font> )
 ;<br/>

<a NAME="E3:44"/>
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  <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font></span>)</span>,<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#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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/>

<b>reconsider </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c6">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> ,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:44_1"/><i><font color="Green" title="E44">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> holds  <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">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="Olive" title="b1">p</font></span>)</span></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="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <font color="Maroon" title="c4">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:44"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="mesfunc2.html#T6" title="MESFUNC2:th.6">MESFUNC2:6</a></i>;<br/>
<a NAME="E4:44_1"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</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="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:44"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E3:44_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="mesfunc5.html#T24" target="_self" title="MESFUNC5:th.24">Th24</a></i>;<br/>
<a NAME="E5:44_1"/>
 ex <font color="Olive" title="b1">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font> st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b1">G</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T5" title="MESFUNC1:th.5">MESFUNC1:5</a>, <a class="ref" href="mesfunc2.html#T5" title="MESFUNC2:th.5">MESFUNC2:5</a></i>;<br/>
<a NAME="E6:44_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T52" title="MEASURE1:th.52">MEASURE1:52</a></i>;<br/>
<b>hence </b><a NAME="E7:44_1"/>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</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">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E4:44_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="measure1.html#D9" title="MEASURE1:def.9">MEASURE1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <font color="Maroon" title="c4">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17" title="MESFUNC1:def.17">MESFUNC1:def 17</a></i>;<br/>


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