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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  holds <br/>(  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b3">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b3">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b3">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">E</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  holds <br/>(  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b2">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b2">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b2">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">E</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  holds <br/>(  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Olive" title="b1">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">E</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  implies (  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> ) )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">XX</font> = <font color="Maroon" title="c1">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T7" title="MEASURE1:th.7">MEASURE1:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font>;<br/>
<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div><span class="lab"><font color="Green" title="E27">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/><a NAME="E2:39_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E28">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_2"/>
 not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/><a NAME="E2:39_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/><a NAME="E3:39_1_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:39_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:39_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">hence </span><a NAME="E4:39_1"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/>
<a NAME="E4:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<a NAME="E5:39"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E6:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:39_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="funct_3.html#T37" title="FUNCT_3:th.37">FUNCT_3:37</a></span>;<br/><a NAME="E3:39_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:39"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T29" title="MESFUNC2:th.29">MESFUNC2:29</a></span>;<br/>
<a NAME="E9:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
 <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<a NAME="E10:39"/><span class="kw">then </span>
 <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#T30" title="MESFUNC1:th.30">MESFUNC1:30</a></span>;<br/>
<a NAME="E11:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="mesfunc5.html#T42" title="MESFUNC5:th.42">MESFUNC5:42</a>, <a class="ref" href="mesfunc5.html#T87" title="MESFUNC5:th.87">MESFUNC5:87</a></span>;<br/>
<a NAME="E12:39"/>
<font color="Maroon" title="c5">XX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="mesfunc2.html#D3" title="MESFUNC2:def.3">MESFUNC2:def 3</a></span>;<br/>
<a NAME="E13:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="mesfunc2.html#T26" title="MESFUNC2:th.26">MESFUNC2:26</a>, <a class="ref" href="mesfunc5.html#T87" title="MESFUNC5:th.87">MESFUNC5:87</a></span>;<br/>
<a NAME="E14:39"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E15:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E16:39"/>
<font color="Maroon" title="c4">E</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></span>;<br/>
<a NAME="E17:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>;<br/>
<a NAME="E18:39"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T12" title="MESFUNC2:th.12">MESFUNC2:12</a></span>;<br/>
<a NAME="E19:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
 <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T52" title="SUPINF_2:th.52">SUPINF_2:52</a></span>;<br/>
<a NAME="E20:39"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p4">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">XX</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="mesfunc5.html#T81" title="MESFUNC5:th.81">MESFUNC5:81</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E21:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A19</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:39"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E42">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:39_3"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:39_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/><a NAME="E3:39_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_3"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="mesfunc6.html#T2" title="MESFUNC6:th.2">MESFUNC6:2</a></span>;<br/>
<a NAME="E24:39"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p3">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E21:39"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="mesfunc5.html#T15" title="MESFUNC5:th.15">MESFUNC5:15</a>, <a class="ref" href="mesfunc5.html#T77" title="MESFUNC5:th.77">MESFUNC5:77</a></span>;<br/>
<a NAME="E25:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A22</font></span>: 
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p5">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E22:39"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="mesfunc5.html#T104" title="MESFUNC5:th.104">MESFUNC5:104</a></span>;<br/>
<a NAME="E26:39"/>
 <a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="mesfunc1.html#T30" title="MESFUNC1:th.30">MESFUNC1:30</a></span>;<br/>
<a NAME="E27:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="mesfunc5.html#T42" title="MESFUNC5:th.42">MESFUNC5:42</a></span>;<br/>
<a NAME="E28:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<a NAME="E29:39"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="mesfunc5.html#T15" title="MESFUNC5:th.15">MESFUNC5:15</a>, <a class="ref" href="supinf_2.html#T52" title="SUPINF_2:th.52">SUPINF_2:52</a></span>;<br/>
<a NAME="E30:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<a NAME="E31:39"/>
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<a NAME="E32:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A25</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:39"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E30:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="mesfunc5.html#T88" title="MESFUNC5:th.88">MESFUNC5:88</a></span>;<br/>
<span class="kw">assume </span><a NAME="E33:39"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> )</span><br/>

<a NAME="E34:39"/><span class="kw">then </span>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="xxreal_3.html#T81" title="XXREAL_3:th.81">XXREAL_3:81</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:39"/>
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font> )</span><br/>

<a NAME="E36:39"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="xxreal_3.html#T15" title="XXREAL_3:th.15">XXREAL_3:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:39"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T81" title="XXREAL_3:th.81">XXREAL_3:81</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font></span><br/>

<a NAME="E38:39"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" title="MESFUNC2:func.1">max+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T23" target="_self" title="MESFUNC7:th.23">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:39"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E21:39"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E32:39"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="xxreal_3.html#T81" title="XXREAL_3:th.81">XXREAL_3:81</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
