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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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="b2">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="b3">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b5">P</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>  st <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b5">P</font> &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> ) &amp; <font color="Olive" title="b5">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b3">M</font> &amp; <font color="Olive" title="b5">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b7">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b5">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b6">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b6">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b7">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b6">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Olive" title="b1">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b6">I</font> <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="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Olive" title="b1">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b7">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="b7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">E</font> holds <br/><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b7">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b6">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font></span>)</span>) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b4">P</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>  st <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">P</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> ) &amp; <font color="Olive" title="b4">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b4">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b5">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b5">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b5">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b5">I</font> <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="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b6">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="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b6">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b5">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b3">P</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>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">P</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) &amp; <font color="Olive" title="b3">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b3">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b4">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b4">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b4">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b4">I</font> <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="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b5">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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b5">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b4">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b2">P</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>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">P</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; <font color="Olive" title="b2">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b2">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b3">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b3">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b3">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b4">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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b3">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b3">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</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>  st <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">P</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> ) &amp; <font color="Olive" title="b1">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b1">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Olive" title="b1">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b2">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b2">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b3">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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b2">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> ) &amp; <font color="Maroon" title="c6">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c6">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; (  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><br/>  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></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> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) &amp;  <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c6">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b2">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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E5:17"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c6">P</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">and </span><br/><a NAME="E6:17"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 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><br/>  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></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> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E7:17"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 <a href="mesfunc1.html#K20" title="MESFUNC1:func.20">eq_dom</a> (<font color="Maroon" title="c6">P</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b2">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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )</span><br/>

<a NAME="E8:17"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D7" title="MESFUNC8:def.7">MESFUNC8:def 7</a></span>;<br/>
<a NAME="E9:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K6</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K7</a>(<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>)) = <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E11:17"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></span>
<a NAME="E12:17"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 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><br/>  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/>(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <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="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/>(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> implies (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<a NAME="E2:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<a NAME="E3:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/>
<a NAME="E4:17_1"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>;<br/>
<a NAME="E5:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/>
<a NAME="E6:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T12" title="EXTREAL2:th.12">EXTREAL2:12</a></span>;<br/>
<a NAME="E7:17_1"/>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="extreal2.html#T12" title="EXTREAL2:th.12">EXTREAL2:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:17_1"/>
(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E29">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E1:17_2_1"/>
<font color="Maroon" title="c9">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E2:17_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T8" title="RINFSUP2:th.8">RINFSUP2:8</a></span>;<br/><a NAME="E3:17_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_2_1"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_2_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:17_2"/><span class="kw">then </span>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun10.html#T5" target="_self" title="MESFUN10:th.5">Th5</a></span>;<br/><a NAME="E4:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="mesfunc8.html#D7" title="MESFUNC8:def.7">MESFUNC8:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">y</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> implies  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font></span><br/><a NAME="E2:17_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_2_2"/>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:17_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></span>;<br/><a NAME="E7:17_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D4" title="XXREAL_2:def.4">XXREAL_2:def 4</a></span>;<br/><a NAME="E8:17_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T47" title="NAT_1:th.47">NAT_1:47</a></span>;<br/><a NAME="E9:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T27" title="RINFSUP2:th.27">RINFSUP2:27</a></span>;<br/><a NAME="E10:17_2"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K3" title="RINFSUP2:func.3">inferior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T23" title="RINFSUP2:th.23">RINFSUP2:23</a></span>;<br/><a NAME="E11:17_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:17_2"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="mesfunc8.html#D7" title="MESFUNC8:def.7">MESFUNC8:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E13:17_2"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="extreal2.html#T12" title="EXTREAL2:th.12">EXTREAL2:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:17"/>
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="mesfunc8.html#T24" title="MESFUNC8:th.24">MESFUNC8:24</a></span>;<br/>
<a NAME="E15:17"/><span class="kw">then </span>
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="mesfunc5.html#T102" title="MESFUNC5:th.102">MESFUNC5:102</a></span>;<br/>
<a NAME="E16:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A21</font></span>: 
 ex <font color="Olive" title="b1">E3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span></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="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E3</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E3</font></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T109" title="MESFUNC5:th.109">MESFUNC5:109</a></span>;<br/>
<a NAME="E17:17"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<a NAME="E18:17"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<a NAME="E19:17"/>
 <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<a NAME="E20:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A24</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>) is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:17"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">assume </span><a NAME="E1:17_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A25</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/><a NAME="E3:17_3"/>
 <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="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="supinf_2.html#T51" title="SUPINF_2:th.51">SUPINF_2:51</a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_3"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> is   <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></span><br/><a NAME="E1:17_4"/>
<font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E27">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_4"/>
<font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> is   <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E36">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><br/><a NAME="E1:17_5"/>
<font color="Maroon" title="c8">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_5"/>
<font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E27">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c7">G</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E23:17"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="seqfunc.html#T1" title="SEQFUNC:th.1">SEQFUNC:1</a></span>;<br/>
<a NAME="E26:17"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="mesfunc5.html#T30" title="MESFUNC5:th.30">MESFUNC5:30</a></span>;<br/>
<a NAME="E27:17"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font></span><br/>

<a NAME="E1:17_6"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:17"><span class="lab"><font color="Green" title="E36">A28</font></span></a></span>;<br/>
<a NAME="E2:17_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E3:17_6"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_6"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E26:17"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E22:17"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E2:17_6"><span class="lab"><font color="Green" title="E39">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E39">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c10">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><br/><span class="kw">thus </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span> = 
<font color="Maroon" title="c5">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K6</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K7</a>(<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>)) = <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">H</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E30:17"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c8">G</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:17"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<a NAME="E32:17"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>;<br/>
<a NAME="E33:17"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font></span><br/>

<a NAME="E1:17_8"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_8_1"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span><br/><a NAME="E2:17_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>;<br/><a NAME="E3:17_8_1"/>
<span class="p1"><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="mesfunc1.html#K11" title="MESFUNC1:func.11">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:17_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E1:17_8_1"><span class="lab"><font color="Green" title="E45">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_8_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_8_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:17_8"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_8"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E2:17_8"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="mesfunc5.html#T102" title="MESFUNC5:th.102">MESFUNC5:102</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )</span><br/><a NAME="E1:17_9"/><span class="lab"><font color="Green" title="E44">A41</font></span>: 
<font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:17"><span class="lab"><font color="Green" title="E36">A28</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_9_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> implies  <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">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_9_1"/><span class="lab"><font color="Green" title="E45">A42</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:17_9_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="txt" href="#E1:17_9"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E1:17_9_1"><span class="lab"><font color="Green" title="E45">A42</font></span></a></span>;<br/><a NAME="E4:17_9_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/><a NAME="E5:17_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A43</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></span>;<br/><a NAME="E6:17_9_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T7" title="XXREAL_3:th.7">XXREAL_3:7</a></span>;<br/><a NAME="E7:17_9_1"/><span class="kw">then </span>
 <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">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_9_1"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/><span class="kw">hence </span><a NAME="E8:17_9_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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_9_1"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:17_9"/>
<font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_9"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="supinf_2.html#T52" title="SUPINF_2:th.52">SUPINF_2:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/><a NAME="E4:17_9"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/><a NAME="E5:17_9"/><span class="kw">then </span>
<font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_9"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="mesfunc5.html#T108" title="MESFUNC5:th.108">MESFUNC5:108</a></span>;<br/><a NAME="E6:17_9"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">E1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font></span>)</span> &amp; <font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E7:17_9"/>
<font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E35:17"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17"><span class="lab"><font color="Green" title="E38">A30</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">IG</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><span class="kw"> such that </span><br/><a NAME="E37:17"/><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c11">IG</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E38:17"/><span class="lab"><font color="Green" title="E45">A45</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c10">G</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c11">IG</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:17"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="ref" href="mesfun10.html#T7" target="_self" title="MESFUN10:th.7">Th7</a></span>;<br/>
<a NAME="E39:17"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <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>  =  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E41:17"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c12">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">I</font> as   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> ;<br/>
<a NAME="E43:17"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
 <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<a NAME="E44:17"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>) &amp; <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>) &amp; <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )</span><br/>

<a NAME="E1:17_10"/>
<font color="Maroon" title="c14">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:17_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A50</font></span>: 
<font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:17"><span class="lab"><font color="Green" title="E47">A47</font></span></a></span>;<br/>
<a NAME="E3:17_10"/><span class="lab"><font color="Green" title="E51">A51</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/>
<a NAME="E4:17_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A52</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<a NAME="E5:17_10"/>
 <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_10"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_10"/>
( <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>) &amp; <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_10"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E4:17_10"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">IG</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span><br/><a NAME="E1:17_11"/><span class="lab"><font color="Green" title="E50">A53</font></span>: 
<font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:17"><span class="lab"><font color="Green" title="E36">A28</font></span></a></span>;<br/><a NAME="E2:17_11"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/><a NAME="E3:17_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A54</font></span>: 
 ex <font color="Olive" title="b1">E2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E2</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E2</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_11"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="ref" href="mesfunc5.html#T109" title="MESFUNC5:th.109">MESFUNC5:109</a></span>;<br/><a NAME="E4:17_11"/><span class="lab"><font color="Green" title="E52">A55</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/><a NAME="E5:17_11"/><span class="lab"><font color="Green" title="E53">A56</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/><a NAME="E6:17_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/><a NAME="E7:17_11"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E44:17"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E3:17_11"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="txt" href="#E5:17_11"><span class="lab"><font color="Green" title="E53">A56</font></span></a>, <a class="txt" href="#E4:17_11"><span class="lab"><font color="Green" title="E52">A55</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:17_11"/>
<font color="Maroon" title="c11">IG</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:17"><span class="lab"><font color="Green" title="E44">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E46:17"/><span class="kw">then </span>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c11">IG</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:17"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="ref" href="mesfun10.html#T10" target="_self" title="MESFUN10:th.10">Th10</a></span>;<br/>
<a NAME="E47:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E24:17"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E26:17"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E22:17"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E38:17"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="txt" href="#E16:17"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E17:17"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E32:17"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="mesfun10.html#T11" target="_self" title="MESFUN10:th.11">Th11</a></span>;<br/>
<a NAME="E48:17"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</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"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:17"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E39:17"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="ref" href="xxreal_3.html#T56" title="XXREAL_3:th.56">XXREAL_3:56</a></span>;<br/>
<a NAME="E49:17"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:17"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E39:17"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="ref" href="xxreal_3.html#T30" title="XXREAL_3:th.30">XXREAL_3:30</a></span>;<br/>
<a NAME="E50:17"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T7" title="XXREAL_3:th.7">XXREAL_3:7</a></span>;<br/>
<a NAME="E51:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A57</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_12"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is   <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></span><br/><a NAME="E1:17_12"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:17"><span class="lab"><font color="Green" title="E40">A33</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_12"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is   <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E52">A59</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_13"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span></span><br/><a NAME="E1:17_13"/>
<font color="Maroon" title="c14">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_13"/>
<font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:17"><span class="lab"><font color="Green" title="E40">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E54:17"/><span class="lab"><font color="Green" title="E53">A60</font></span>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E54">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <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#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_14"/><span class="lab"><font color="Green" title="E55">A62</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <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#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span><br/><a NAME="E2:17_14"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><br/>

<a NAME="E1:17_14_1"/>
<font color="Maroon" title="c15">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:17_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A63</font></span>: 
<span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T8" title="RINFSUP2:th.8">RINFSUP2:8</a></span>;<br/>
<a NAME="E3:17_14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:17"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:17_14"><span class="lab"><font color="Green" title="E55">A62</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_14_1"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_14_1"><span class="lab"><font color="Green" title="E56">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:17_14"/><span class="kw">then </span>
 <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun10.html#T4" target="_self" title="MESFUN10:th.4">Th4</a></span>;<br/><a NAME="E4:17_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A64</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_14"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_14_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">y</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> implies <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_14_2"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">y</font></span><br/><a NAME="E2:17_14_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_14_2"/>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:17"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E1:17_14"><span class="lab"><font color="Green" title="E55">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:17_14"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a></span>;<br/><a NAME="E7:17_14"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K17" title="SUPINF_2:func.17">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D3" title="XXREAL_2:def.3">XXREAL_2:def 3</a></span>;<br/><a NAME="E8:17_14"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T47" title="NAT_1:th.47">NAT_1:47</a></span>;<br/><a NAME="E9:17_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A65</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T27" title="RINFSUP2:th.27">RINFSUP2:27</a></span>;<br/><a NAME="E10:17_14"/>
<span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K4" title="RINFSUP2:func.4">superior_realsequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T23" title="RINFSUP2:th.23">RINFSUP2:23</a></span>;<br/><a NAME="E11:17_14"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_14"><span class="lab"><font color="Green" title="E57">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:17_14"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_14"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E13:17_14"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <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#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_14"><span class="lab"><font color="Green" title="E56">A64</font></span></a>, <a class="ref" href="extreal2.html#T12" title="EXTREAL2:th.12">EXTREAL2:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E56:17"/>
 <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="mesfunc8.html#T23" title="MESFUNC8:th.23">MESFUNC8:23</a></span>;<br/>
<a NAME="E57:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A66</font></span>: 
 <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E54:17"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E55:17"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="mesfunc5.html#T102" title="MESFUNC5:th.102">MESFUNC5:102</a></span>;<br/>
<a NAME="E58:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A67</font></span>: 
 ex <font color="Olive" title="b1">E6</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></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="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E6</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p4">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E6</font></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfun10.html#T13" target="_self" title="MESFUN10:th.13">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">E1</font> =  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>);<br/>
<a NAME="E59:17"/><span class="lab"><font color="Green" title="E57">A68</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:17"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<a NAME="E60:17"/>
 <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:17"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">e1</font> =  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:17"><span class="lab"><font color="Green" title="E57">A68</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/>
<a NAME="E62:17"/><span class="lab"><font color="Green" title="E58">A69</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c15">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T2" title="SUPINF_2:th.2">SUPINF_2:2</a></span>;<br/>
<a NAME="E63:17"/><span class="lab"><font color="Green" title="E59">A70</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K6</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K7</a>(<font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>)) =  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E65:17"/><span class="lab"><font color="Green" title="E60">A71</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c16">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c17">H</font> = <font color="Maroon" title="c9">H</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:17"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E52:17"><span class="lab"><font color="Green" title="E51">A58</font></span></a>, <a class="ref" href="seqfunc.html#T1" title="SEQFUNC:th.1">SEQFUNC:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E61">A72</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_15"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font></span><br/><a NAME="E1:17_15"/><span class="lab"><font color="Green" title="E62">A73</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:17"><span class="lab"><font color="Green" title="E52">A59</font></span></a></span>;<br/><a NAME="E2:17_15"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/><a NAME="E3:17_15"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">E</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E26:17"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E22:17"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E1:17_15"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_15"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_16"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span><br/><span class="kw">thus </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> = 
<font color="Maroon" title="c5">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:17"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:17"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">H</font> = <font color="Maroon" title="c17">H</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<a NAME="E70:17"/><span class="lab"><font color="Green" title="E62">A74</font></span>: 
 <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E63">A75</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_17"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )</span><br/><a NAME="E1:17_17"/><span class="lab"><font color="Green" title="E64">A76</font></span>: 
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:17"><span class="lab"><font color="Green" title="E52">A59</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_17_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font></span><br/><a NAME="E2:17_17_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/><a NAME="E3:17_17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:17_17_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">P</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:17_17"/>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_17"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="ref" href="mesfunc7.html#T1" title="MESFUNC7:th.1">MESFUNC7:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/><a NAME="E4:17_17"/>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/><a NAME="E5:17_17"/><span class="kw">then </span>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_17"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="ref" href="mesfun10.html#T12" target="_self" title="MESFUN10:th.12">Th13</a></span>;<br/><a NAME="E6:17_17"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">E4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font></span>)</span> &amp; <font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D17" title="MESFUNC5:def.17">MESFUNC5:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E7:17_17"/>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:17"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E72:17"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:17"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">IH</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><span class="kw"> such that </span><br/><a NAME="E74:17"/><span class="lab"><font color="Green" title="E64">A77</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c19">IH</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E75:17"/><span class="lab"><font color="Green" title="E65">A78</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c18">H</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c19">IH</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:17"><span class="lab"><font color="Green" title="E63">A75</font></span></a>, <a class="ref" href="mesfun10.html#T7" target="_self" title="MESFUN10:th.7">Th7</a></span>;<br/>
<a NAME="E76:17"/><span class="lab"><font color="Green" title="E66">A79</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_18"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> is   <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></span><br/><a NAME="E1:17_18"/>
<font color="Maroon" title="c16">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:17"><span class="lab"><font color="Green" title="E60">A71</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_18"/>
<font color="Maroon" title="c16">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> is   <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">F1</font> = <font color="Maroon" title="c16">F1</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:17"><span class="lab"><font color="Green" title="E60">A71</font></span></a>, <a class="ref" href="seqfunc.html#T1" title="SEQFUNC:th.1">SEQFUNC:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E67">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_19"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span></span><br/><a NAME="E1:17_19"/>
<font color="Maroon" title="c21">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_19"/>
<font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:17"><span class="lab"><font color="Green" title="E60">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_20"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font>, <font color="Maroon" title="c22">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font></span>)</span></span><br/><a NAME="E1:17_20"/>
<font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:17"><span class="lab"><font color="Green" title="E67">A80</font></span></a></span>;<br/><a NAME="E2:17_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A81</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></span>;<br/><a NAME="E3:17_20"/>
<font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:17"><span class="lab"><font color="Green" title="E67">A80</font></span></a></span>;<br/><a NAME="E4:17_20"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_20"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_20"><span class="lab"><font color="Green" title="E68">A81</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">F1</font> = <font color="Maroon" title="c20">F1</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/>
<a NAME="E82:17"/><span class="lab"><font color="Green" title="E68">A82</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</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="c4">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T9" title="MESFUNC2:th.9">MESFUNC2:9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E69">A83</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_21"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span><br/><a NAME="E1:17_21"/>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:17"><span class="lab"><font color="Green" title="E52">A59</font></span></a></span>;<br/><a NAME="E2:17_21"/><span class="kw">then </span>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T8" title="MESFUNC2:th.8">MESFUNC2:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_21"/>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:17"><span class="lab"><font color="Green" title="E67">A80</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E84:17"/>
<font color="Maroon" title="c21">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:17"><span class="lab"><font color="Green" title="E67">A80</font></span></a></span>;<br/>
<a NAME="E85:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A84</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E86:17"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F1</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E67:17"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>;<br/>
<a NAME="E87:17"/><span class="kw">then </span>
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c18">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c21">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:17"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E22:17"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E83:17"><span class="lab"><font color="Green" title="E69">A83</font></span></a>, <a class="ref" href="mesfun10.html#T11" target="_self" title="MESFUN10:th.11">Th11</a></span>;<br/>
<a NAME="E88:17"/><span class="kw">then </span>
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c18">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:17"><span class="lab"><font color="Green" title="E67">A80</font></span></a>, <a class="txt" href="#E85:17"><span class="lab"><font color="Green" title="E70">A84</font></span></a>, <a class="ref" href="mesfun10.html#T15" target="_self" title="MESFUN10:th.15">Th16</a></span>;<br/>
<a NAME="E89:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A85</font></span>: 
 <a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c18">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T8" title="MESFUNC2:th.8">MESFUNC2:8</a></span>;<br/>
<a NAME="E90:17"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a></span>;<br/>
<a NAME="E91:17"/><span class="kw">then </span>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E92:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A86</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>5</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <span class="kw">-&gt; </span>   <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>  =  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">I1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E94:17"/><span class="lab"><font color="Green" title="E73">A87</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c22">I1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c23">I1</font> = <font color="Maroon" title="c22">I1</font> as   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> ;<br/>
<div><span class="lab"><font color="Green" title="E74">A88</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_22"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">I1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><br/><a NAME="E1:17_22"/>
<font color="Maroon" title="c24">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:17_22"/>
<font color="Maroon" title="c23">I1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:17"><span class="lab"><font color="Green" title="E73">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E97:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A89</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c23">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun10.html#T14" target="_self" title="MESFUN10:th.14">Th15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_23"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">IH</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">I1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><br/><a NAME="E1:17_23"/><span class="lab"><font color="Green" title="E76">A90</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17"><span class="lab"><font color="Green" title="E42">A35</font></span></a></span>;<br/><a NAME="E2:17_23"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</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="c4">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T9" title="MESFUNC2:th.9">MESFUNC2:9</a></span>;<br/><a NAME="E3:17_23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A91</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_23"><span class="lab"><font color="Green" title="E76">A90</font></span></a>, <a class="ref" href="mesfunc5.html#T110" title="MESFUNC5:th.110">MESFUNC5:110</a></span>;<br/><a NAME="E4:17_23"/><span class="lab"><font color="Green" title="E78">A92</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></span>;<br/><a NAME="E5:17_23"/><span class="kw">then </span>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" title="MESFUNC1:def.7">MESFUNC1:def 7</a></span>;<br/><a NAME="E6:17_23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A93</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/><a NAME="E7:17_23"/>
<font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:17"><span class="lab"><font color="Green" title="E52">A59</font></span></a></span>;<br/><a NAME="E8:17_23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A94</font></span>: 
 ex <font color="Olive" title="b1">E5</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">E5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">P</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</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="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E5</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K8" title="MESFUNC1:func.8">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E5</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:17_23"><span class="lab"><font color="Green" title="E76">A90</font></span></a>, <a class="ref" href="mesfun10.html#T13" target="_self" title="MESFUN10:th.13">Th14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">In</font> = <font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:17"><span class="lab"><font color="Green" title="E49">A49</font></span></a></span>;<br/><a NAME="E10:17_23"/><span class="lab"><font color="Green" title="E81">A95</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c25">In</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T1" title="EXTREAL1:th.1">EXTREAL1:1</a></span>;<br/><a NAME="E11:17_23"/><span class="lab"><font color="Green" title="E82">A96</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c25">In</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T2" title="SUPINF_2:th.2">SUPINF_2:2</a></span>;<br/><a NAME="E12:17_23"/>
<font color="Maroon" title="c6">P</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/><a NAME="E13:17_23"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c24">n</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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E44:17"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E8:17_23"><span class="lab"><font color="Green" title="E80">A94</font></span></a>, <a class="txt" href="#E4:17_23"><span class="lab"><font color="Green" title="E78">A92</font></span></a>, <a class="txt" href="#E6:17_23"><span class="lab"><font color="Green" title="E79">A93</font></span></a>, <a class="txt" href="#E3:17_23"><span class="lab"><font color="Green" title="E77">A91</font></span></a>, <a class="txt" href="#E10:17_23"><span class="lab"><font color="Green" title="E81">A95</font></span></a>, <a class="txt" href="#E11:17_23"><span class="lab"><font color="Green" title="E82">A96</font></span></a></span>;<br/><a NAME="E14:17_23"/><span class="kw">then </span>
<font color="Maroon" title="c19">IH</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:17"><span class="lab"><font color="Green" title="E64">A77</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:17_23"/>
<font color="Maroon" title="c19">IH</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">I1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:17"><span class="lab"><font color="Green" title="E74">A88</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E99:17"/><span class="kw">then </span>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c19">IH</font> <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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c23">I1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:17"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="ref" href="mesfun10.html#T10" target="_self" title="MESFUN10:th.10">Th10</a></span>;<br/>
<a NAME="E100:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p3">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E54:17"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E75:17"><span class="lab"><font color="Green" title="E65">A78</font></span></a>, <a class="txt" href="#E97:17"><span class="lab"><font color="Green" title="E75">A89</font></span></a>, <a class="txt" href="#E89:17"><span class="lab"><font color="Green" title="E71">A85</font></span></a>, <a class="txt" href="#E57:17"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="txt" href="#E58:17"><span class="lab"><font color="Green" title="E56">A67</font></span></a>, <a class="txt" href="#E63:17"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E92:17"><span class="lab"><font color="Green" title="E72">A86</font></span></a>, <a class="txt" href="#E82:17"><span class="lab"><font color="Green" title="E68">A82</font></span></a>, <a class="ref" href="mesfunc5.html#T110" title="MESFUNC5:th.110">MESFUNC5:110</a></span>;<br/>
<a NAME="E101:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</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"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p3">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:17"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="txt" href="#E76:17"><span class="lab"><font color="Green" title="E66">A79</font></span></a>, <a class="ref" href="xxreal_3.html#T56" title="XXREAL_3:th.56">XXREAL_3:56</a></span>;<br/>
<a NAME="E102:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</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#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c6">P</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:17"><span class="lab"><font color="Green" title="E62">A74</font></span></a>, <a class="txt" href="#E76:17"><span class="lab"><font color="Green" title="E66">A79</font></span></a>, <a class="ref" href="xxreal_3.html#T30" title="XXREAL_3:th.30">XXREAL_3:30</a></span>;<br/>
<a NAME="E103:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A97</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</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#K2" title="SUPINF_2:func.2">-</a> <span class="p2">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T7" title="XXREAL_3:th.7">XXREAL_3:7</a></span>;<br/>
<a NAME="E104:17"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T1" title="EXTREAL1:th.1">EXTREAL1:1</a></span>;<br/>
<a NAME="E105:17"/><span class="kw">then </span>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:17"><span class="lab"><font color="Green" title="E76">A97</font></span></a>, <a class="txt" href="#E62:17"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<a NAME="E106:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A98</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T38" title="XXREAL_3:th.38">XXREAL_3:38</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_24"><span class="kw">now </span></a><div class="add"><a NAME="E1:17_24"/><span class="lab"><font color="Green" title="E78">A99</font></span>: 
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T39" title="RINFSUP2:th.39">RINFSUP2:39</a></span>;<br/><a NAME="E2:17_24"/><span class="lab"><font color="Green" title="E79">A100</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D9" title="MESFUNC8:def.9">MESFUNC8:def 9</a></span>;<br/><span class="kw">assume </span><a NAME="E3:17_24"/><span class="lab"><font color="Green" title="E80">A101</font></span>: 
 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="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) )</span><br/><a NAME="E4:17_24"/><span class="lab"><font color="Green" title="E81">A102</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</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="17_24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_24_1"/><span class="lab"><font color="Green" title="E82">A103</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<a NAME="E2:17_24_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c24">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:17_24"><span class="lab"><font color="Green" title="E80">A101</font></span></a>, <a class="txt" href="#E2:17_24"><span class="lab"><font color="Green" title="E79">A100</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17_24_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_24_1"><span class="lab"><font color="Green" title="E82">A103</font></span></a>, <a class="ref" href="mesfunc8.html#T14" title="MESFUNC8:th.14">MESFUNC8:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:17_24"/><span class="lab"><font color="Green" title="E82">A104</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</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="17_24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_24_2"/><span class="lab"><font color="Green" title="E83">A105</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<a NAME="E2:17_24_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c24">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:17_24"><span class="lab"><font color="Green" title="E80">A101</font></span></a>, <a class="txt" href="#E2:17_24"><span class="lab"><font color="Green" title="E79">A100</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17_24_2"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_24_2"><span class="lab"><font color="Green" title="E83">A105</font></span></a>, <a class="ref" href="mesfunc8.html#T14" title="MESFUNC8:th.14">MESFUNC8:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:17_24"/><span class="lab"><font color="Green" title="E83">A106</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_24"><span class="lab"><font color="Green" title="E79">A100</font></span></a>, <a class="ref" href="mesfunc8.html#D8" title="MESFUNC8:def.8">MESFUNC8:def 8</a></span>;<br/><a NAME="E7:17_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A107</font></span>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_24"><span class="lab"><font color="Green" title="E82">A104</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/><a NAME="E8:17_24"/><span class="lab"><font color="Green" title="E85">A108</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_24"><span class="lab"><font color="Green" title="E79">A100</font></span></a>, <a class="ref" href="mesfunc8.html#D7" title="MESFUNC8:def.7">MESFUNC8:def 7</a></span>;<br/><a NAME="E9:17_24"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:17"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E4:17_24"><span class="lab"><font color="Green" title="E81">A102</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/><a NAME="E10:17_24"/><span class="kw">then </span>
 <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:17"><span class="lab"><font color="Green" title="E77">A98</font></span></a>, <a class="txt" href="#E7:17_24"><span class="lab"><font color="Green" title="E84">A107</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:17_24"/><span class="kw">then </span>
 <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_24"><span class="lab"><font color="Green" title="E78">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:17_24"/><span class="lab"><font color="Green" title="E86">A109</font></span>: 
<font color="Maroon" title="c13">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T40" title="RINFSUP2:th.40">RINFSUP2:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>)</span><br/><a NAME="E13:17_24"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T41" title="RINFSUP2:th.41">RINFSUP2:41</a></span>;<br/><a NAME="E14:17_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A110</font></span>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:17"><span class="lab"><font color="Green" title="E77">A98</font></span></a>, <a class="txt" href="#E6:17_24"><span class="lab"><font color="Green" title="E83">A106</font></span></a>, <a class="txt" href="#E5:17_24"><span class="lab"><font color="Green" title="E82">A104</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/><a NAME="E15:17_24"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_24"><span class="lab"><font color="Green" title="E86">A109</font></span></a>, <a class="ref" href="rinfsup2.html#T41" title="RINFSUP2:th.41">RINFSUP2:41</a></span>;<br/><a NAME="E16:17_24"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:17"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E8:17_24"><span class="lab"><font color="Green" title="E85">A108</font></span></a>, <a class="txt" href="#E4:17_24"><span class="lab"><font color="Green" title="E81">A102</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E17:17_24"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c13">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17_24"><span class="lab"><font color="Green" title="E87">A110</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E108:17"/>
 ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">n</font> <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="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp;  <a href="rinfsup2.html#K6" title="RINFSUP2:func.6">lim_inf</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K5" title="MESFUNC8:func.5">lim_inf</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp;  <a href="rinfsup2.html#K5" title="RINFSUP2:func.5">lim_sup</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K6" title="MESFUNC8:func.6">lim_sup</a> <font color="Maroon" title="c2">F</font></span>)</span>) &amp; ( (  for <font color="Olive" title="b2">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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  ) implies ( <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <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="c4">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c2">F</font></span>)</span>) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:17"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E51:17"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E106:17"><span class="lab"><font color="Green" title="E77">A98</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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