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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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><br/>  for <font color="Olive" title="b4">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b2">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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>  &amp; <font color="Olive" title="b4">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b4">f</font> is <font color="Olive" title="b3">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b4">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><br/>  for <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b1">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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>  &amp; <font color="Olive" title="b3">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b3">f</font> is <font color="Olive" title="b2">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b3">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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>  &amp; <font color="Olive" title="b2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b2">f</font> is <font color="Olive" title="b1">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <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> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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>  &amp; <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b1">f</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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>  &amp; <font color="Maroon" title="c5">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c5">f</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  implies <font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<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> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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> 
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">f</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E5:47"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<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"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="mesfunc5.html#D7" onmouseover="rs('mesfunc5/D7')" onmouseout="rh()">MESFUNC5:def 7</a></span>;<br/>
<a NAME="E6:47"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="mesfunc6.html#D1" onmouseover="rs('mesfunc6/D1')" onmouseout="rh()">MESFUNC6:def 1</a></span>;<br/>
<a NAME="E7:47"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">real-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="integra1.html#T15" onmouseover="rs('integra1/T15')" onmouseout="rh()">INTEGRA1:15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1"/>
( <font color="Maroon" title="c4">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1"/>
<font color="Maroon" title="c4">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><div class="add"><a NAME="E2:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">LB</font> =  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font></span>)</span> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">UB</font> =  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">f</font></span>)</span> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">r</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> =  <a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c4">E</font>,<font color="Maroon" title="c1">X</font>) as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a>, <a class="ref" href="numbers.html#T31" onmouseover="rs('numbers/T31')" onmouseout="rh()">NUMBERS:31</a></span>;<br/><a NAME="E6:47_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E7:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>;<br/><a NAME="E8:47_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="mesfunc7.html#T24" onmouseover="rs('mesfunc7/T24')" onmouseout="rh()">MESFUNC7:24</a></span>;<br/><a NAME="E9:47_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</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="ref" href="mesfunc5.html#T110" onmouseover="rs('mesfunc5/T110')" onmouseout="rh()">MESFUNC5:110</a></span>;<br/><a NAME="E10:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T112" onmouseover="rs('mesfunc5/T112')" onmouseout="rh()">MESFUNC5:112</a></span>;<br/><span class="lab"><font color="Green" title="E51">A11</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"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c4">E</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E12:47_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI: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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></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#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI: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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></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#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_2"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI: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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></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#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E2:47_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D7" onmouseover="rs('mesfunc5/D7')" onmouseout="rh()">MESFUNC5:def 7</a></span>;<br/>
<a NAME="E3:47_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>: 
( <font color="Maroon" title="c6">LB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">UB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T3" onmouseover="rs('xxreal_2/T3')" onmouseout="rh()">XXREAL_2:3</a>, <a class="ref" href="xxreal_2.html#T4" onmouseover="rs('xxreal_2/T4')" onmouseout="rh()">XXREAL_2:4</a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E4:47_1_1_2"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E5:47_1_1_2"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D7" onmouseover="rs('mesfunc5/D7')" onmouseout="rh()">MESFUNC5:def 7</a></span>;<br/>
<a NAME="E6:47_1_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">LB</font> &amp; <font color="Maroon" title="c7">UB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/>
<a NAME="E7:47_1_1_2"/><span class="kw">then </span>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:47_1_1_2"/><span class="kw">then </span>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:47_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T5" onmouseover="rs('absvalue/T5')" onmouseout="rh()">ABSVALUE:5</a></span>;<br/>
<a NAME="E10:47_1_1_2"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">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="#E1:47_1_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></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#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47_1_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_1_2"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="xxreal_3.html#D5" onmouseover="rs('xxreal_3/D5')" onmouseout="rh()">XXREAL_3:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:47_1_1_2"/>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c6">LB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c7">UB</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <font color="Maroon" title="c9">g</font></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#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1_1_2"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E5:47_1_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="extreal1.html#T12" onmouseover="rs('extreal1/T12')" onmouseout="rh()">EXTREAL1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:47_1_1"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</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="#E6:47"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E10:47_1_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E11:47_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="mesfunc5.html#T102" onmouseover="rs('mesfunc5/T102')" onmouseout="rh()">MESFUNC5:102</a></span>;<br/><span class="kw">hence </span><a NAME="E14:47_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><div class="add"><a NAME="E2:47_1_2"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
(  <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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp;  <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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></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="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="mesfunc2.html#D2" onmouseover="rs('mesfunc2/D2')" onmouseout="rh()">MESFUNC2:def 2</a>, <a class="ref" href="mesfunc2.html#D3" onmouseover="rs('mesfunc2/D3')" onmouseout="rh()">MESFUNC2:def 3</a></span>;<br/><a NAME="E3:47_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI: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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> holds <br/><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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="mesfunc2.html#D2" onmouseover="rs('mesfunc2/D2')" onmouseout="rh()">MESFUNC2:def 2</a></span>;<br/><a NAME="E4:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A20</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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="mesfunc2.html#T25" onmouseover="rs('mesfunc2/T25')" onmouseout="rh()">MESFUNC2:25</a>, <a class="ref" href="mesfunc5.html#T87" onmouseover="rs('mesfunc5/T87')" onmouseout="rh()">MESFUNC5:87</a></span>;<br/><a NAME="E5:47_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI: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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> holds <br/><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="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="mesfunc2.html#D3" onmouseover="rs('mesfunc2/D3')" onmouseout="rh()">MESFUNC2:def 3</a></span>;<br/><a NAME="E6:47_1_2"/><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#K2" title="MESFUNC2:func.2">max-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="mesfunc2.html#T26" onmouseover="rs('mesfunc2/T26')" onmouseout="rh()">MESFUNC2:26</a>, <a class="ref" href="mesfunc5.html#T87" onmouseover="rs('mesfunc5/T87')" onmouseout="rh()">MESFUNC5:87</a></span>;<br/><span class="kw">hence </span><a NAME="E7:47_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E4:47_1_2"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="mesfunc5.html#D17" onmouseover="rs('mesfunc5/D17')" onmouseout="rh()">MESFUNC5:def 17</a>, <a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="kw">end;</span></div>

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