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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b5">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="b6">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">F</font> &amp; <font color="Olive" title="b4">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b5">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Olive" title="b4">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp;  ex <font color="Olive" title="b7">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b7">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Olive" title="b6">F</font> &amp; <font color="Olive" title="b7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b6">F</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"><font color="Olive" title="b3">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b4">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</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> <font color="Olive" title="b7">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b4">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">E</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b4">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="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">F</font> &amp; <font color="Olive" title="b3">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Olive" title="b3">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b6">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b6">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Olive" title="b5">F</font> &amp; <font color="Olive" title="b6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b5">F</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"><font color="Olive" title="b2">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b3">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</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> <font color="Olive" title="b6">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b3">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">E</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> <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="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b4">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> &amp; <font color="Olive" title="b2">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Olive" title="b2">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b5">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b5">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Olive" title="b4">F</font> &amp; <font color="Olive" title="b5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b4">F</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"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b2">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></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> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b5">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b2">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</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#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b2">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></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> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b2">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b1">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b2">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c7">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> ) ) )</span><br/>











<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E20">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V5" title="SUPINF_1:attr.5">bounded</a> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> ) )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c8">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E1</font> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c8">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> &amp; <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c8">E1</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E3:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a>, <a class="ref" href="mesfunc2.html#T29" title="MESFUNC2:th.29">MESFUNC2:29</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font> &amp; <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E24">A5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is <a href="supinf_1.html#V3" title="SUPINF_1:attr.3">bounded_above</a> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="supinf_1.html#D13" title="SUPINF_1:def.13">SUPINF_1:def 13</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc7.html#T16" target="_self" title="MESFUNC7:th.16">Th16</a></i>;<br/>
<a NAME="E8:33"/><i><font color="Green" title="E26">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc7.html#T16" target="_self" title="MESFUNC7:th.16">Th16</a>, <a class="ref" href="mesfunc2.html#T35" title="MESFUNC2:th.35">MESFUNC2:35</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
( not  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> &amp; not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T1" title="SUPINF_1:th.1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6" title="SUPINF_1:th.6">SUPINF_1:6</a></i>;<br/>
<a NAME="E10:33"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">k0</font> =  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font>, <font color="Maroon" title="c10">l0</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E6:33"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="supinf_1.html#T83" title="SUPINF_1:th.83">SUPINF_1:83</a>, <a class="ref" href="supinf_1.html#T84" title="SUPINF_1:th.84">SUPINF_1:84</a></i>;<br/>
<a NAME="E12:33"/>
( <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font> &amp; <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font> )
 
<b>by </b><i><a class="ref" href="extreal2.html#T49" title="EXTREAL2:th.49">EXTREAL2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">k1</font> = <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>, <font color="Maroon" title="c12">l1</font> = <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E14:33"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E15:33"/><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E16:33"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c6">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E17:33"/><i><font color="Green" title="E30">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>;<br/>
<a NAME="E18:33"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E19:33"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E20:33"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T2" target="_self" title="MESFUNC7:th.2">Th2</a></i>;<br/>
<a NAME="E21:33"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E22:33"/><b>then </b><i><font color="Green" title="E33">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E23:33"/><i><font color="Green" title="E34">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E24:33"/><i><font color="Green" title="E35">A16</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>;<br/>
<a NAME="E25:33"/><b>then </b><i><font color="Green" title="E36">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E26:33"/><i><font color="Green" title="E37">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E27:33"/><i><font color="Green" title="E38">A19</font></i>: 
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a>, <a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>
<a NAME="E28:33"/><i><font color="Green" title="E39">A20</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> implies <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )</span><br/>

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span></span><br/>

<a NAME="E2:33_1"/><b>then </b><i><font color="Green" title="E40">A21</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E15:33"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">fx</font> = <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc7.html#E8:33"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<a NAME="E4:33_1"/><i><font color="Green" title="E41">A22</font></i>: 
( <font color="Maroon" title="c9">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">fx</font> &amp; <font color="Maroon" title="c14">fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l0</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E27:33"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="mesfunc7.html#E2:33_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a>, <a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E5:33_1"/>
( <font color="Maroon" title="c10">l0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a>, <a class="ref" href="complex1.html#T162" title="COMPLEX1:th.162">COMPLEX1:162</a></i>;<br/>
<a NAME="E6:33_1"/><b>then </b>
( <font color="Maroon" title="c10">l0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> &amp; <font color="Maroon" title="c10">l0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:33_1"/><b>then </b><i><font color="Green" title="E42">A23</font></i>: 
<font color="Maroon" title="c10">l0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:33_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E9:33_1"/><b>then </b><i><font color="Green" title="E43">A24</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></i>;<br/>
<a NAME="E10:33_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k0</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T162" title="COMPLEX1:th.162">COMPLEX1:162</a></i>;<br/>
<a NAME="E11:33_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k0</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E9:33_1"><i><font color="Green" title="E43">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:33_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">fx</font> &amp; <font color="Maroon" title="c14">fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E4:33_1"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="mesfunc7.html#E7:33_1"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:33_1"/><b>then </b><i><font color="Green" title="E44">A25</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c14">fx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E14:33_1"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c14">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">l0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 
<b>by </b><i><a class="ref" href="extreal2.html#T49" title="EXTREAL2:th.49">EXTREAL2:49</a></i>;<br/>
<b>hence </b><a NAME="E15:33_1"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E13:33_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E29:33"/><i><font color="Green" title="E40">A26</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T106" title="MESFUNC5:th.106">MESFUNC5:106</a></i>;<br/>
<a NAME="E30:33"/><b>then </b><i><font color="Green" title="E41">A27</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E31:33"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E29:33"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>
<a NAME="E32:33"/><b>then </b><i><font color="Green" title="E42">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E33:33"/><i><font color="Green" title="E43">A29</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E5:33"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E7:33"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="mesfunc7.html#T15" target="_self" title="MESFUNC7:th.15">Th15</a></i>;<br/>
<a NAME="E34:33"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E17:33"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/>
<a NAME="E35:33"/><b>then </b><i><font color="Green" title="E44">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E33:33"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="mesfunc5.html#T48" title="MESFUNC5:th.48">MESFUNC5:48</a></i>;<br/>
<a NAME="E36:33"/><i><font color="Green" title="E45">A31</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</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"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</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"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E46">A32</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</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"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:33_2"/><b>then </b><i><font color="Green" title="E47">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E3:33_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:33_2"/><b>then </b><i><font color="Green" title="E48">A34</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E5:33_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/>
<a NAME="E6:33_2"/><b>then </b><i><font color="Green" title="E49">A35</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p3">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E28:33"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="txt" href="mesfunc7.html#E4:33_2"><i><font color="Green" title="E48">A34</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>;<br/>
<a NAME="E7:33_2"/><i><font color="Green" title="E50">A36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E22:33"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:33_2"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E17:33"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>;<br/>
<a NAME="E9:33_2"/><b>then </b><i><font color="Green" title="E51">A37</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:33_2"><i><font color="Green" title="E47">A33</font></i></a>, <a class="txt" href="mesfunc7.html#E4:33_2"><i><font color="Green" title="E48">A34</font></i></a>, <a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>;<br/>
<a NAME="E10:33_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E23:33"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E11:33_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_2"><i><font color="Green" title="E46">A32</font></i></a>, <a class="txt" href="mesfunc7.html#E4:33_2"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="mesfunc7.html#E7:33_2"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:33_2"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</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"><font color="Maroon" title="c11">k1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">l1</font></span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:33_2"><i><font color="Green" title="E49">A35</font></i></a>, <a class="txt" href="mesfunc7.html#E9:33_2"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E37:33"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E22:33"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="mesfunc7.html#E32:33"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="mesfunc7.html#E35:33"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="mesfunc5.html#T108" title="MESFUNC5:th.108">MESFUNC5:108</a></i>;<br/>
<a NAME="E38:33"/><b>then </b><i><font color="Green" title="E46">A38</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E35:33"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="mesfunc5.html#T106" title="MESFUNC5:th.106">MESFUNC5:106</a></i>;<br/>
<a NAME="E39:33"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> implies <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></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><br/>

<b>assume </b><a NAME="E1:33_3"/><i><font color="Green" title="E47">A39</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></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><br/>

<a NAME="E2:33_3"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E3:33_3"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T70" title="EXTREAL2:th.70">EXTREAL2:70</a></i>;<br/>
<b>hence </b><a NAME="E4:33_3"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></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> 
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E14:33"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_3"><i><font color="Green" title="E47">A39</font></i></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E40:33"/><b>then </b><i><font color="Green" title="E47">A40</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E41:33"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E42:33"/><b>then </b><i><font color="Green" title="E48">A41</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E5:33"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E7:33"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="mesfunc7.html#E40:33"><i><font color="Green" title="E47">A40</font></i></a>, <a class="ref" href="mesfunc7.html#T15" target="_self" title="MESFUNC7:th.15">Th15</a></i>;<br/>
<a NAME="E43:33"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a></i>;<br/>
<a NAME="E44:33"/><b>then </b><i><font color="Green" title="E49">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E42:33"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="mesfunc5.html#T48" title="MESFUNC5:th.48">MESFUNC5:48</a></i>;<br/>
<a NAME="E45:33"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p5"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p5"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<b>assume </b><a NAME="E1:33_4"/><i><font color="Green" title="E50">A43</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p5"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<b>then </b><span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> = 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p3"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E14:33"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="mesfunc7.html#E15:33"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="mesfunc7.html#E24:33"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>
<br/>.= 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T70" title="EXTREAL2:th.70">EXTREAL2:70</a></i>
;<br/>
<a NAME="E3:33_4"/><b>then </b><i><font color="Green" title="E51">A44</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T56" title="EXTREAL2:th.56">EXTREAL2:56</a></i>;<br/>
<a NAME="E4:33_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E17:33"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E5:33_4"/><b>then </b><i><font color="Green" title="E52">A45</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E6:33_4"/><i><font color="Green" title="E53">A46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:33_4"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:33_4"/><b>then </b>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E19:33"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="mesfunc7.html#E5:33_4"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:33_4"/>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p5"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E17:33"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33_4"><i><font color="Green" title="E51">A44</font></i></a>, <a class="txt" href="mesfunc7.html#E5:33_4"><i><font color="Green" title="E52">A45</font></i></a>, <a class="txt" href="mesfunc7.html#E6:33_4"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E46:33"/><b>then </b><i><font color="Green" title="E50">A47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E19:33"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E38:33"><i><font color="Green" title="E46">A38</font></i></a>, <a class="txt" href="mesfunc7.html#E44:33"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="mesfunc5.html#T108" title="MESFUNC5:th.108">MESFUNC5:108</a></i>;<br/>
<a NAME="E47:33"/><i><font color="Green" title="E51">A48</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> holds <br/>( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> implies ( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> ) )</span><br/>

<b>assume </b><a NAME="E1:33_5"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )</span><br/>

<a NAME="E2:33_5"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E15:33"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:33_5"/><b>then </b><i><font color="Green" title="E52">A49</font></i>: 
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E27:33"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a>, <a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E4:33_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/>
<b>hence </b><a NAME="E5:33_5"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:33_5"><i><font color="Green" title="E52">A49</font></i></a>, <a class="ref" href="extreal1.html#T42" title="EXTREAL1:th.42">EXTREAL1:42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E48:33"/><i><font color="Green" title="E52">A50</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E49:33"/><b>then </b><i><font color="Green" title="E53">A51</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E50:33"/><i><font color="Green" title="E54">A52</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E24:33"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="mesfunc7.html#E48:33"><i><font color="Green" title="E52">A50</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E51:33"/>
(  <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"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> &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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T2" target="_self" title="MESFUNC7:th.2">Th2</a></i>;<br/>
<a NAME="E52:33"/><b>then </b><i><font color="Green" title="E55">A53</font></i>: 
(  <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"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> &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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E30:33"><i><font color="Green" title="E41">A27</font></i></a>, <a class="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>
<a NAME="E53:33"/>
<font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E29:33"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>
<a NAME="E54:33"/><b>then </b><i><font color="Green" title="E56">A54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E55:33"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<b>assume </b><a NAME="E1:33_6"/><i><font color="Green" title="E57">A55</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:33_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E48:33"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:33_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_6"><i><font color="Green" title="E57">A55</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:33_6"/><b>then </b><i><font color="Green" title="E58">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E47:33"><i><font color="Green" title="E51">A48</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_6"><i><font color="Green" title="E57">A55</font></i></a></i>;<br/>
<a NAME="E5:33_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_6"><i><font color="Green" title="E57">A55</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:33_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_6"><i><font color="Green" title="E57">A55</font></i></a>, <a class="txt" href="mesfunc7.html#E4:33_6"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E56:33"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T1" target="_self" title="MESFUNC7:th.1">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">V1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E58:33"/><i><font color="Green" title="E57">A57</font></i>: 
( <font color="Maroon" title="c13">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> &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"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">V1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">V1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E46:33"><i><font color="Green" title="E50">A47</font></i></a>, <a class="txt" href="mesfunc7.html#E54:33"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="mesfunc7.html#T3" target="_self" title="MESFUNC7:th.3">Th3</a></i>;<br/>
<a NAME="E59:33"/><i><font color="Green" title="E58">A58</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">V1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">V1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E58:33"><i><font color="Green" title="E57">A57</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E60:33"/>
<font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E29:33"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="mesfunc5.html#T116" title="MESFUNC5:th.116">MESFUNC5:116</a></i>;<br/>
<a NAME="E61:33"/><b>then </b><i><font color="Green" title="E59">A59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T103" title="MESFUNC5:th.103">MESFUNC5:103</a></i>;<br/>
<a NAME="E62:33"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<b>assume </b><a NAME="E1:33_7"/><i><font color="Green" title="E60">A60</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span><br/>

<a NAME="E2:33_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <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> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E50:33"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="mesfunc1.html#D5" title="MESFUNC1:def.5">MESFUNC1:def 5</a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>
<a NAME="E3:33_7"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <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> <font color="Maroon" title="c10">l0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E16:33"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_7"><i><font color="Green" title="E60">A60</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:33_7"/><b>then </b><i><font color="Green" title="E61">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E47:33"><i><font color="Green" title="E51">A48</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_7"><i><font color="Green" title="E60">A60</font></i></a></i>;<br/>
<a NAME="E5:33_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_7"><i><font color="Green" title="E60">A60</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E6:33_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <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"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_7"><i><font color="Green" title="E60">A60</font></i></a>, <a class="txt" href="mesfunc7.html#E4:33_7"><i><font color="Green" title="E61">A61</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E63:33"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T1" target="_self" title="MESFUNC7:th.1">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">V2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E65:33"/><i><font color="Green" title="E60">A62</font></i>: 
( <font color="Maroon" title="c14">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> &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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">V2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">V2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E46:33"><i><font color="Green" title="E50">A47</font></i></a>, <a class="txt" href="mesfunc7.html#E61:33"><i><font color="Green" title="E59">A59</font></i></a>, <a class="ref" href="mesfunc7.html#T3" target="_self" title="MESFUNC7:th.3">Th3</a></i>;<br/>
<a NAME="E66:33"/><i><font color="Green" title="E61">A63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">V2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">V2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E25:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="mesfunc7.html#E26:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E49:33"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="mesfunc7.html#E65:33"><i><font color="Green" title="E60">A62</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E67:33"/><i><font color="Green" title="E62">A64</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> &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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></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>  )
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E46:33"><i><font color="Green" title="E50">A47</font></i></a>, <a class="ref" href="mesfunc5.html#T102" title="MESFUNC5:th.102">MESFUNC5:102</a></i>;<br/>
<a NAME="E68:33"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_8_1"/>
(  <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or  <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_8_1_1"><b>suppose </b></a><a NAME="E1:33_8_1_1"/><i><font color="Green" title="E63">A65</font></i>: 
 <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )</span><br/><div class="add"><a NAME="E2:33_8_1_1"/><b>then </b><i><font color="Green" title="E64">A66</font></i>: 
 <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc7.html#E4:33"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="mesfunc5.html#T98" title="MESFUNC5:th.98">MESFUNC5:98</a></i>;<br/><a NAME="E3:33_8_1_1"/><i><font color="Green" title="E65">A67</font></i>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> &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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></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>  )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E30:33"><i><font color="Green" title="E41">A27</font></i></a>, <a class="ref" href="mesfunc5.html#T102" title="MESFUNC5:th.102">MESFUNC5:102</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">c1</font> =  <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1" title="EXTREAL1:th.1">EXTREAL1:1</a></i>;<br/><a NAME="E5:33_8_1_1"/><i><font color="Green" title="E66">A68</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></i>;<br/><a NAME="E6:33_8_1_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font> )
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E7:33_8_1_1"/><b>then </b><i><font color="Green" title="E67">A69</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></i>;<br/><a NAME="E8:33_8_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">k0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">c1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">l0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">c1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l0</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">c1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font></span>)</span> )
 ;<br/><a NAME="E9:33_8_1_1"/><b>then </b><i><font color="Green" title="E68">A70</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E5:33_8_1_1"><i><font color="Green" title="E66">A68</font></i></a>, <a class="txt" href="mesfunc7.html#E7:33_8_1_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E10:33_8_1_1"/>
( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">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="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">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="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33_8_1_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="ref" href="extreal1.html#T34" title="EXTREAL1:th.34">EXTREAL1:34</a></i>;<br/><a NAME="E11:33_8_1_1"/><b>then </b><i><font color="Green" title="E69">A71</font></i>: 
( <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="ref" href="extreal2.html#T4" title="EXTREAL2:th.4">EXTREAL2:4</a></i>;<br/><b>set </b><font color="Maroon" title="c16">c2</font> = <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c17">c3</font> =  <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="mesfunc7.html#E67:33"><i><font color="Green" title="E62">A64</font></i></a>, <a class="ref" href="extreal1.html#T1" title="EXTREAL1:th.1">EXTREAL1:1</a></i>;<br/><a NAME="E13:33_8_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c3</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">c</font> = <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  ;<br/><a NAME="E15:33_8_1_1"/><i><font color="Green" title="E70">A72</font></i>: 
( <font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c18">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E52:33"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="mesfunc7.html#E58:33"><i><font color="Green" title="E57">A57</font></i></a>, <a class="txt" href="mesfunc7.html#E59:33"><i><font color="Green" title="E58">A58</font></i></a>, <a class="txt" href="mesfunc7.html#E65:33"><i><font color="Green" title="E60">A62</font></i></a>, <a class="txt" href="mesfunc7.html#E66:33"><i><font color="Green" title="E61">A63</font></i></a>, <a class="txt" href="mesfunc7.html#E2:33_8_1_1"><i><font color="Green" title="E64">A66</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33_8_1_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="txt" href="mesfunc7.html#E9:33_8_1_1"><i><font color="Green" title="E68">A70</font></i></a>, <a class="txt" href="mesfunc7.html#E11:33_8_1_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="ref" href="extreal1.html#T48" title="EXTREAL1:th.48">EXTREAL1:48</a></i>;<br/><a NAME="E16:33_8_1_1"/>
( <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c3</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="ref" href="extreal1.html#T32" title="EXTREAL1:th.32">EXTREAL1:32</a></i>;<br/><a NAME="E17:33_8_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p5"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c3</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font></span>)</span> &amp; <font color="Maroon" title="c17">c3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">c1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c15">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><span class="p4"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/><a NAME="E18:33_8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c18">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_1"><i><font color="Green" title="E63">A65</font></i></a>, <a class="txt" href="mesfunc7.html#E3:33_8_1_1"><i><font color="Green" title="E65">A67</font></i></a>, <a class="ref" href="extreal2.html#T19" title="EXTREAL2:th.19">EXTREAL2:19</a></i>;<br/><b>hence </b><a NAME="E19:33_8_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E15:33_8_1_1"><i><font color="Green" title="E70">A72</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_8_1_2"><b>suppose </b></a><a NAME="E1:33_8_1_2"/><i><font color="Green" title="E63">A73</font></i>: 
 <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"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_8_1_2"/><i><font color="Green" title="E64">A74</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="mesfunc7.html#E3:33_8_1_2"><i><font color="Green" title="E64">A74</font></i></a></i>;<br/><a NAME="E5:33_8_1_2"/>
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:33_8_1_2"><i><font color="Green" title="E64">A74</font></i></a>, <a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/><a NAME="E6:33_8_1_2"/><b>then </b><i><font color="Green" title="E65">A75</font></i>: 
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k0</font> &amp; <font color="Maroon" title="c9">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_8_1_2"/>
 <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>  <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E52:33"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="mesfunc7.html#E58:33"><i><font color="Green" title="E57">A57</font></i></a>, <a class="txt" href="mesfunc7.html#E59:33"><i><font color="Green" title="E58">A58</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_8_1_2"><i><font color="Green" title="E63">A73</font></i></a>, <a class="ref" href="extreal1.html#T22" title="EXTREAL1:th.22">EXTREAL1:22</a></i>;<br/><a NAME="E8:33_8_1_2"/><b>then </b><i><font color="Green" title="E66">A76</font></i>: 
 <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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc7.html#E52:33"><i><font color="Green" title="E55">A53</font></i></a>, <a class="txt" href="mesfunc7.html#E65:33"><i><font color="Green" title="E60">A62</font></i></a>, <a class="txt" href="mesfunc7.html#E66:33"><i><font color="Green" title="E61">A63</font></i></a>, <a class="txt" href="mesfunc7.html#E1:33_8_1_2"><i><font color="Green" title="E63">A73</font></i></a>, <a class="ref" href="extreal1.html#T22" title="EXTREAL1:th.22">EXTREAL1:22</a></i>;<br/><a NAME="E9:33_8_1_2"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c9">k0</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:33_8_1_2"><i><font color="Green" title="E63">A73</font></i></a>, <a class="ref" href="extreal1.html#T22" title="EXTREAL1:th.22">EXTREAL1:22</a></i>;<br/><b>hence </b><a NAME="E10:33_8_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E6:33_8_1_2"><i><font color="Green" title="E65">A75</font></i></a>, <a class="txt" href="mesfunc7.html#E8:33_8_1_2"><i><font color="Green" title="E66">A76</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E69:33"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &amp;  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">F</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"><font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#K13" title="MESFUNC1:func.13">(#)</a> <span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</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> <font color="Olive" title="b1">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c5">g</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">E</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E18:33"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E22:33"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="mesfunc7.html#E32:33"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="mesfunc7.html#E35:33"><i><font color="Green" title="E44">A30</font></i></a>, <a class="txt" href="mesfunc7.html#E36:33"><i><font color="Green" title="E45">A31</font></i></a>, <a class="ref" href="mesfunc5.html#T108" title="MESFUNC5:th.108">MESFUNC5:108</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
