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<span class="kw">let </span><font color="Maroon" title="c1">Omega</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">f</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b3">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>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b2">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/><font color="Olive" title="b2">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Olive" title="b3">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Olive" title="b1">Sigma</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">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>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Olive" title="b2">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c3">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  holds <br/><font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Olive" title="b1">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c3">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  implies <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c3">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font></span><br/>

<a NAME="E2:41"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:41"/>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>;<br/>
<a NAME="E4:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font><span class="kw"> such that </span><br/><a NAME="E6:41"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font>
 <span class="kw">and </span><br/><a NAME="E7:41"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D2" target="_self" title="RANDOM_1:def.2">Def2</a></span>;<br/>
<a NAME="E8:41"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:41"/>
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c4">r</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="random_1.html#D2" target="_self" title="RANDOM_1:def.2">Def2</a>, <a class="ref" href="mesfun6c.html#T29" title="MESFUN6C:th.29">MESFUN6C:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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