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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">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<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>assume </b><a NAME="E1:21"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E39</font></i>: 
( ( for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K3" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> ) implies for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K3" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font>
 ;<br/><div><b>hereby </b>
<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="E1:21_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</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="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K3" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17">great_dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</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></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:21"/>
( <font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">A</font> iff  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:21"/>
( <font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">A</font> iff for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K3" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#T33">MESFUNC1:33</a>, , </i>;<br/>


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