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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">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</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>assume </b><a NAME="E1:140"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:140"/><b>then </b><i><font color="Green">E37</font></i>: 
(  ex <font color="Olive">A</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">A</font> ) &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:140"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:140"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a> </span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K2">-infty</a> </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#T37">MESFUNC1:37</a>, <a class="ref" href="mesfunc1.html#T38">MESFUNC1:38</a></i>;<br/>
<a NAME="E6:140"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a>  implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K2">-infty</a>  implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D16">MESFUNC1:def 16</a></i>;<br/>
<a NAME="E7:140"/><b>then </b><i><font color="Green">E40</font></i>: 
(  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K2">-infty</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:140"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">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="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K2">-infty</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K19">eq_dom</a> <font color="Maroon">f</font>,<a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E9:140"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E7:140"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:140"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="mesfunc1.html#D11">MESFUNC1:def 11</a></i>;<br/>
<a NAME="E13:140"/><i><font color="Green">E43</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">C1</font> = <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font>;<br/>
<b>set </b><font color="Maroon">C2</font> = <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font>;<br/>
<a NAME="E14:140"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E15:140"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E7:140"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E16:140"/><b>then </b>
( <font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E17:140"/><b>then </b><i><font color="Green">E94</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E18:140"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E19:140"/><b>then </b><i><font color="Green">E95</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<a NAME="E20:140"/><i><font color="Green">E98</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E21:140"/><b>then </b><i><font color="Green">E99</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font> &amp;  <a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E22:140"/>
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E23:140"/><b>then </b><i><font color="Green">E100</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E24:140"/>
( <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">B2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T106">XBOOLE_1:106</a></i>;<br/>
<a NAME="E25:140"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B1</font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="measure1.html#K4">\</a> <font color="Maroon">B2</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E26:140"/><b>then </b><i><font color="Green">E101</font></i>: 
(  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:140_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:140_1"/><b>then </b><i><font color="Green">E104</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:140_1"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:140_1_1"/><i><font color="Green">E107</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  =  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>;<br/>
<a NAME="E2:140_1_1"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:140_1_1"/><i><font color="Green">E109</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) ) ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S3">PARTFUN1:sch 3</a>(<a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>);<br/></i>
<a NAME="E5:140_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:140_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:140_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:140_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E9:140_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:140_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E2:140_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E3:140_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:140_1_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E5:140_1_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:140_1_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:140_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:140_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>
<a NAME="E5:140_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E6:140_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:140_1_1_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:140_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<a NAME="E7:140_1_1_1"/>
 <a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T32">MESFUNC2:32</a></i>;<br/>
<a NAME="E8:140_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B1</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:140_1_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc1.html#T41">MESFUNC1:41</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:140_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, </i>;<br/>
<a NAME="E11:140_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<a NAME="E12:140_1_1"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:140_1_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:140_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:140_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:140_1_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/>
<a NAME="E5:140_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/>
<a NAME="E6:140_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc2.html#D2">MESFUNC2:def 2</a></i>;<br/>
<a NAME="E7:140_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:140_1_1_2"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:140_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:140_1_1"/><b>then </b>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , , , </i>;<br/>
<a NAME="E15:140_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E16:140_1_1"/><i><font color="Green">E137</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, , , <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="txt" href="mesfunc5.html#E9:140_1_1"><i><font color="Green">E123</font></i></a></i>;<br/>
<a NAME="E17:140_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, , <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, </i>;<br/>
<b>hence </b><a NAME="E18:140_1_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , <a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="txt" href="mesfunc5.html#E27:140"><i><font color="Green">E102</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:140_1_2"/>
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1_2_1"><b>suppose </b></a><a NAME="E1:140_1_2_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:140_1_2_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E3:140_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1_2_2"><b>suppose </b></a><a NAME="E1:140_1_2_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:140_1_2_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E3:140_1_2_2"/><b>then </b><i><font color="Green">E140</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/><b>reconsider </b><font color="Maroon">MB</font> = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">I</font> =  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:140_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E7:140_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E8:140_1_2_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E9:140_1_2_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E10:140_1_2_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:140"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E11:140_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="xreal_1.html#T157">XREAL_1:157</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:140_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:140_2"/><i><font color="Green">E184</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:140_2"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/><span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:140_2_1"/><i><font color="Green">E186</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  =  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>;<br/>
<a NAME="E2:140_2_1"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:140_2_1"/><i><font color="Green">E188</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) ) ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S3">PARTFUN1:sch 3</a>(<a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>);<br/></i>
<a NAME="E5:140_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:140_2_1"/><b>then </b><i><font color="Green">E189</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:140_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E8:140_2_1"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E9:140_2_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:140_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E2:140_2_1_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E3:140_2_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:140_2_1_1"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E5:140_2_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:140_2_1_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:140_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green">E196</font></i>: <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, , , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>
<a NAME="E4:140_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E5:140_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/>
<b>hence </b><a NAME="E6:140_2_1_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:140_2_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="mesfunc1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<a NAME="E7:140_2_1_1"/>
 <a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T32">MESFUNC2:32</a></i>;<br/>
<a NAME="E8:140_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3">chi</a> <font color="Maroon">B2</font>,<font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:140_2_1_1"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc1.html#T41">MESFUNC1:41</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:140_2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E11:140_2_1"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>
<a NAME="E12:140_2_1"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:140_2_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:140_2_1_2"/><b>then </b><i><font color="Green">E201</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:140_2_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/>
<a NAME="E4:140_2_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/>
<a NAME="E5:140_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E6:140_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a>, , <a class="ref" href="mesfunc2.html#D3">MESFUNC2:def 3</a></i>;<br/>
<a NAME="E7:140_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:140_2_1_2"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:140_2_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:140_2_1"/><b>then </b>
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">B2</font></span>)</span>
 
<b>by </b><i>, , , , , , </i>;<br/>
<a NAME="E15:140_2_1"/><b>then </b><i><font color="Green">E204</font></i>: 
 <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>
<a NAME="E16:140_2_1"/><i><font color="Green">E205</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , </i>;<br/>
<a NAME="E17:140_2_1"/>
<font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, , , </i>;<br/>
<b>hence </b><a NAME="E18:140_2_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, , , , <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="txt" href="mesfunc5.html#E27:140"><i><font color="Green">E102</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:140_2_2"/>
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_2_2_1"><b>suppose </b></a><a NAME="E1:140_2_2_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:140_2_2_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:140_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_2_2_2"><b>suppose </b></a><a NAME="E1:140_2_2_2"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:140_2_2_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:140_2_2_2"/><b>then </b><i><font color="Green">E208</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, <a class="txt" href="mesfunc5.html#E1:140_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/><b>reconsider </b><font color="Maroon">MB</font> = <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a>, <a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a></i>;<br/><b>reconsider </b><font color="Maroon">I</font> =  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E6:140_2_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E7:140_2_2_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, <a class="txt" href="mesfunc5.html#E1:140_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E8:140_2_2_2"/><b>then </b><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:140_2_2_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">MB</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">MB</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E10:140_2_2_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">I</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:140_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E11:140_2_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="xreal_1.html#T157">XREAL_1:157</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E29:140"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T41">PROB_1:41</a></i>;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B1</font> as    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="measure1.html#D13">MEASURE1:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span>

<br/>.= 
0
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:140"><i><font color="Green">E142</font></i></a>, <a class="ref" href="measure1.html#T71">MEASURE1:71</a></i>
;<br/>


</div>
