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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  ex <font color="Olive" title="b4">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font> st <br/>( <font color="Olive" title="b4">E</font> <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="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> ) holds <br/><font color="Olive" title="b3">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  ex <font color="Olive" title="b3">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b3">E</font> <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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> ) holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  ex <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">E</font> <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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</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#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp;  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <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="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) implies <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:8"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">and </span><br/><a NAME="E4:8"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">E</font> <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="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E6:8"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c5">E</font> <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="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E7:8"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/>
<a NAME="E8:8"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/>
<a NAME="E9:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A8</font></span>: 
(  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">V47</a>() &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T31" title="MESFUNC2:th.31">MESFUNC2:31</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="lab"><font color="Green" title="E16">A9</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c5">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E17">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:8_2"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><a NAME="E2:8_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E1:8_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T92" title="XXREAL_3:th.92">XXREAL_3:92</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:8_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:8"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="mesfunc6.html#D6" title="MESFUNC6:def.6">MESFUNC6:def 6</a></span>;<br/>
<a NAME="E13:8"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></span>;<br/>
<a NAME="E14:8"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E15:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E16:8"/>
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="mesfunc7.html#T24" title="MESFUNC7:th.24">MESFUNC7:24</a></span>;<br/>
<a NAME="E17:8"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="mesfunc7.html#T17" title="MESFUNC7:th.17">MESFUNC7:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:8"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="mesfunc6.html#D9" title="MESFUNC6:def.9">MESFUNC6:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
