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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="numbers.html#K6">ExtREAL</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><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#R2" target="_self">is_measurable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</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" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:51_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#T2" target="_self">Th2</a>, <a class="ref" href="mesfunc1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E4:51"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E2:51_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:51"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="mesfunc1.html#D2" target="_self">Def2</a>);<br/></i>
<a NAME="E7:51"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" target="_self">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure2.html#K1">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc1.html#E4:51"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:51"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" target="_self">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a> </span>)</span> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D1" target="_self">Def1</a></i>;<br/>


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