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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> , <font color="Maroon">g</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="E1:145"/><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> &amp; <font color="Maroon">g</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 ;<br/>

<b>consider </b><font color="Maroon">E0</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:145"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">E0</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">E0</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, </i>;<br/>
<a NAME="E4:145"/><i><font color="Green">E38</font></i>: 
 ex <font color="Olive">Gf</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">Gf</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">Gf</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">NF</font> = <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> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E40:145"><i><font color="Green">E124</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">NG</font> = <span class="p1">(<span class="default"><font color="Maroon">g</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">g</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> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E40:145"><i><font color="Green">E124</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">NFG</font> = <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>;<br/>
<a NAME="E7:145"/>
( <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">NF</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">NG</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E40:145"><i><font color="Green">E124</font></i></a>, </i>;<br/>
<a NAME="E8:145"/><b>then </b>
( <font color="Maroon">NF</font> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font> &amp; <font color="Maroon">NG</font> is    <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/>
<a NAME="E9:145"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T70">MEASURE1:70</a></i>;<br/>
<a NAME="E10:145"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="measure1.html#D13">MEASURE1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">E</font> = <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span>;<br/>
<a NAME="E11:145"/>
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> <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> implies <font color="Olive">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="Olive">x</font> <a href="hidden.html#R2">in</a> <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> implies <font color="Olive">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="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</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> implies <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> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</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> implies <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> ) )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E12:145"/><b>then </b>
( <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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:145"/><b>then </b>
( <font color="Maroon">NF</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">NG</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:145"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E15:145"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><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="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4"><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#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p4"><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></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span>;<br/>
<a NAME="E16:145"/>
( <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="tarski.html#R1">c=</a> <font color="Maroon">NF</font> &amp; <font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">NG</font> &amp; <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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</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="tarski.html#R1">c=</a> <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> &amp; <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#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">g</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> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E17:145"/><b>then </b>
( <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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">NF</font> &amp; <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#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">NG</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:145"/><b>then </b>
<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#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</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></span>)</span> <a href="subset_1.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#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">g</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E19:145"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_1.html#T34">XBOOLE_1:34</a>, <a class="txt" href="mesfunc5.html#E47:145"><i><font color="Green">E136</font></i></a></i>;<br/>
<a NAME="E20:145"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E9:145"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">DFPG</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">NFPG</font> = <font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span>;<br/>
<a NAME="E22:145"/>
<font color="Maroon">DFPG</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E0</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, <a class="txt" href="mesfunc5.html#E47:145"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E23:145"/><b>then </b>
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E24:145"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">E0</font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>;<br/>
<a NAME="E25:145"/>
<font color="Maroon">E0</font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:145"/><b>then </b>
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E51:145"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E27:145"/><b>then </b>
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E44:145"><i><font color="Green">E133</font></i></a>, <a class="ref" href="measure1.html#T69">MEASURE1:69</a></i>;<br/>
<a NAME="E28:145"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="measure1.html#D13">MEASURE1:def 13</a></i>;<br/>
<a NAME="E29:145"/>
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DFPG</font> <a href="measure1.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>;<br/>
<a NAME="E30:145"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E48:145"><i><font color="Green">E137</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E31:145"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">DFPG</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E32:145"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E33:145"/><b>then </b><i><font color="Green">E100</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, , <a class="ref" href="mesfunc1.html#T34">MESFUNC1:34</a></i>;<br/>
<a NAME="E34:145"/><i><font color="Green">E101</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</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> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</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> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</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="E35:145"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E56:145"><i><font color="Green">E143</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E36:145"/><b>then </b><i><font color="Green">E103</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , </i>;<br/>
<a NAME="E37:145"/><i><font color="Green">E104</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></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="145_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:145_1"/><i><font color="Green">E106</font></i>: 
<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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:145_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">f</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>, <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E4:145_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</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="E5:145_1"/><b>then </b><i><font color="Green">E107</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:145_1"/><b>then </b><i><font color="Green">E108</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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> 
 ;<br/><a NAME="E2:145_1_1"/><b>then </b>
<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="funct_1.html#T68">FUNCT_1:68</a>, </i>;<br/><a NAME="E3:145_1_1"/><b>then </b>
<font color="Maroon">ff</font> <a href="supinf_2.html#K13">.</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:145_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ff</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>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:145_1_1"/>
<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="tarski.html#R1">c=</a> <font color="Maroon">NF</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:145_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><a NAME="E2:145_1_2"/><b>then </b>
<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#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, </i>;<br/><a NAME="E3:145_1_2"/><b>then </b>
<font color="Maroon">ff</font> <a href="supinf_2.html#K13">.</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:145_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ff</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>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:145_1_2"/>
<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="tarski.html#R1">c=</a> <font color="Maroon">NF</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:145_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:145_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:145"/><b>then </b>
( ( 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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 
;<br/>
<a NAME="E39:145"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V4" target="_self">without+infty</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E40:145"/><i><font color="Green">E124</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/>(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:145_2"/><i><font color="Green">E125</font></i>: 
<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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:145_2"/>
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#R2">in</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>, <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E4:145_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</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="E5:145_2"/><b>then </b><i><font color="Green">E126</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:145_2"/><b>then </b><i><font color="Green">E127</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><i><font color="Green">E128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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> 
 ;<br/><a NAME="E2:145_2_1"/><b>then </b>
<font color="Maroon">g</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="funct_1.html#T68">FUNCT_1:68</a>, </i>;<br/><a NAME="E3:145_2_1"/><b>then </b>
<font color="Maroon">gg</font> <a href="supinf_2.html#K13">.</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:145_2_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">gg</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>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:145_2_1"/>
<font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">NG</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:145_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><a NAME="E2:145_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, </i>;<br/><a NAME="E3:145_2_2"/><b>then </b>
<font color="Maroon">gg</font> <a href="supinf_2.html#K13">.</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:145_2_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">gg</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>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:145_2_2"/>
<font color="Maroon">g</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="tarski.html#R1">c=</a> <font color="Maroon">NG</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:145_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:145_2"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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#T36" target="_self">Th36</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:145"/><b>then </b>
( ( 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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 
;<br/>
<a NAME="E42:145"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V4" target="_self">without+infty</a> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_3"><b>now </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="E1:145_3"/>
<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">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:145_3"/><b>then </b><i><font color="Green">E132</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <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#T35" target="_self">Th35</a></i>;<br/><a NAME="E3:145_3"/>
 <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:145_3"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></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="extreal2.html#T59">EXTREAL2:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E44:145"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_4"><b>now </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="E1:145_4"/>
<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">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:145_4"/><b>then </b><i><font color="Green">E134</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:145_4"/>
 <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:145_4"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></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="extreal2.html#T59">EXTREAL2:59</a></i>;<br/></div><b>end;</b></div>
<a NAME="E46:145"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E47:145"/><b>then </b><i><font color="Green">E136</font></i>: 
( <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> <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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E48:145"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E9:145"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="mesfunc2.html#T7">MESFUNC2:7</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E48:145"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E49:145"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E50:145"/>
( ( 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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <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> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a>, </i>;<br/>
<a NAME="E51:145"/><b>then </b><i><font color="Green">E139</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span> is <a href="mesfunc5.html#V3" target="_self">without-infty</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E52:145"/><b>then </b>
(  <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E53:145"/><b>then </b><i><font color="Green">E140</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <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="E54:145"/><i><font color="Green">E141</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a>, </i>;<br/>
<a NAME="E55:145"/><i><font color="Green">E142</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></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="E56:145"/><i><font color="Green">E143</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E57:145"/><i><font color="Green">E184</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E58:145"/><i><font color="Green">E185</font></i>: 
 <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<a NAME="E59:145"/><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="mesfunc5.html#E1:145"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E60:145"/><i><font color="Green">E187</font></i>: 
(  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp;  <a href="mesfunc2.html#K1">max+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc2.html#T27">MESFUNC2:27</a></i>;<br/>
<a NAME="E61:145"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></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"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E62:145"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="mesfunc5.html#T8" target="_self">Th8</a></i>;<br/>
<b>then </b><i><font color="Green">E188</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"><span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E36:145"><i><font color="Green">E103</font></i></a>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E36:145"><i><font color="Green">E103</font></i></a>, <a class="ref" href="mesfunc5.html#T39" target="_self">Th39</a>, , , , <a class="txt" href="mesfunc5.html#E49:145"><i><font color="Green">E138</font></i></a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>
;<br/>
<a NAME="E64:145"/><i><font color="Green">E189</font></i>: 
( 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"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &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"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E65:145"/><b>then </b><i><font color="Green">E190</font></i>: 
(  <a href="supinf_2.html#K3">-</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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3">-</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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/>
<a NAME="E66:145"/>
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> &amp;  <a href="mesfunc2.html#K2">max-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="txt" href="mesfunc5.html#E46"><i><font color="Green">Lemma157</font></i></a>, , <a class="ref" href="mesfunc2.html#T28">MESFUNC2:28</a></i>;<br/>
<b>then </b> <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"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E36:145"><i><font color="Green">E103</font></i></a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E36:145"><i><font color="Green">E103</font></i></a>, <a class="ref" href="mesfunc5.html#T39" target="_self">Th39</a>, , , , <a class="txt" href="mesfunc5.html#E49:145"><i><font color="Green">E138</font></i></a>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>
;<br/>
<a NAME="E68:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T40" target="_self">Th40</a>, , <a class="ref" href="extreal1.html#T11">EXTREAL1:11</a></i>;<br/>
<a NAME="E69:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T11">EXTREAL1:11</a></i>;<br/>
<a NAME="E70:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/>
<a NAME="E71:145"/><b>then </b>
<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"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E72:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/>
<a NAME="E73:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p5">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p4">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p5">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/>
<a NAME="E74:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a>, , <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/>
<a NAME="E75:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/>
<a NAME="E76:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K2">max-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></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"><a href="mesfunc2.html#K1">max+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p5">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E77:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/>
<a NAME="E78:145"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T41" target="_self">Th41</a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>;<br/>
<a NAME="E79:145"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p2">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p4">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p1">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K1">max+</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K6" target="_self">integral+</a> <font color="Maroon">M</font>,<span class="p3">(<span class="default"><a href="mesfunc2.html#K2">max-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p0">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>;<br/>
<a NAME="E80:145"/><b>then </b><i><font color="Green">E191</font></i>: 
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p4">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E81:145"/><i><font color="Green">E192</font></i>: 
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a></i>;<br/>
<a NAME="E82:145"/>
 <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">E0</font> <a href="measure1.html#K4">\</a> <span class="p3">(<span class="default"><font color="Maroon">NF</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">NG</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, , , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a></i>;<br/>
<b>hence </b><a NAME="E83:145"/>
 ex <font color="Olive">E</font>, <font color="Olive">NFG</font>, <font color="Olive">NFPG</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">E</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">NFG</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">NFG</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc2.html#R1">is_finite</a>  &amp; <font color="Olive">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp; <font color="Olive">E</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">NFG</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">NFG</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc2.html#R1">is_finite</a>  &amp; <font color="Olive">E</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp;  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp; <font color="Olive">E</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Olive">NFPG</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Olive">E</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"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">NFPG</font> &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">NFG</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">NFPG</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">E</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"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Olive">E</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="mesfunc5.html#R1" target="_self">is_integrable_on</a> <font color="Maroon">M</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> &amp;  <a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" target="_self">Integral</a> <font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <font color="Olive">E</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E56:145"><i><font color="Green">E143</font></i></a>, <a class="txt" href="mesfunc5.html#E46:145"><i><font color="Green">E135</font></i></a>, , <a class="txt" href="mesfunc5.html#E42:145"><i><font color="Green">E131</font></i></a>, , , , <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a>, <a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="mesfunc5.html#T43" target="_self">Th43</a>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="txt" href="mesfunc5.html#E48:145"><i><font color="Green">E137</font></i></a>, <a class="txt" href="mesfunc5.html#E55:145"><i><font color="Green">E142</font></i></a>, <a class="txt" href="mesfunc5.html#E49:145"><i><font color="Green">E138</font></i></a>, <a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="txt" href="mesfunc5.html#E54:145"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E53:145"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="mesfunc5.html#T42" target="_self">Th42</a></i>;<br/>


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