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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">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/>



<a NAME="E1:28"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font> implies for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Olive" title="b1">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>;<br/>

<a NAME="E2:28_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:28_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D11" target="_self" title="MESFUNC1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1"/>
<font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28"/>
( ( for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Olive" title="b1">M</font> ) implies <font color="Maroon" title="c2">A</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/><i><font color="Green" title="E23">A3</font></i>: 
for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Olive" title="b1">M</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">M</font> being    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>;<br/>
<a NAME="E3:28_2"/>
<font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E1:28_2"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="measure1.html#D12" title="MEASURE1:def.12">MEASURE1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2"/>
<font color="Maroon" title="c2">A</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">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="E3:28"/>
( <font color="Maroon" title="c2">A</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font> iff for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c3">S</font> holds  <font color="Maroon" title="c2">A</font> <a href="measure1.html#R2" title="MEASURE1:pred.2">is_measurable</a> <font color="Olive" title="b1">M</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:28"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/>


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