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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b5">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#K2" title="NUMBERS:func.2">COMPLEX</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="Olive" title="b5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font> is <font color="Olive" title="b3">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b4">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b7">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="b4">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font> ) &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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b5">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b6">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) holds <br/> <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b5">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b4">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#K2" title="NUMBERS:func.2">COMPLEX</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="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> is <font color="Olive" title="b2">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b3">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="b3">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> ) &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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Olive" title="b4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b5">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) holds <br/> <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b4">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">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#K2" title="NUMBERS:func.2">COMPLEX</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="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b1">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b2">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="b2">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> ) &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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Olive" title="b3">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b4">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) holds <br/> <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b3">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">P</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Olive" title="b1">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="b1">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> ) &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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Olive" title="b2">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b3">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) holds <br/> <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b2">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c5">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</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="b3">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> ) &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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Olive" title="b1">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) holds <br/> <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Olive" title="b1">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a>  ) &amp; <font color="Maroon" title="c5">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> &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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> ) &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>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) implies  <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E3:57"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfun6c.html#V1" title="MESFUN6C:attr.1">-measurable</a> 
 <span class="kw">and </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E5:57"/><span class="lab"><font color="Green" title="E45">A5</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E6:57"/><span class="lab"><font color="Green" title="E46">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>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E7:57"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c4">E</font> <a href="mesfunc6.html#V1" title="MESFUNC6:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="mesfun9c.html#E42"><span class="lab"><font color="Green" title="E34">Lm2</font></span></a></span>;<br/>
<a NAME="E8:57"/>
 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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/>( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/>( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies ( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c8">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:57_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D5" onmouseover="rs('comseq_3/D5')" onmouseout="rh()">COMSEQ_3:def 5</a></span>;<br/>
<a NAME="E3:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T13" onmouseover="rs('comseq_3/T13')" onmouseout="rh()">COMSEQ_3:13</a></span>;<br/>
<a NAME="E4:57_1"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D6" onmouseover="rs('comseq_3/D6')" onmouseout="rh()">COMSEQ_3:def 6</a></span>;<br/>
<a NAME="E5:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T13" onmouseover="rs('comseq_3/T13')" onmouseout="rh()">COMSEQ_3:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:57_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<a NAME="E7:57_1"/><span class="kw">then </span>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/>
<a NAME="E8:57_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</a></span>;<br/>
<a NAME="E9:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun7c.html#D9" onmouseover="rs('mesfun7c/D9')" onmouseout="rh()">MESFUN7C:def 9</a></span>;<br/>
<a NAME="E10:57_1"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E11:57_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">n1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqfunc.html#D10" onmouseover="rs('seqfunc/D10')" onmouseout="rh()">SEQFUNC:def 10</a></span>;<br/>
<a NAME="E13:57_1"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E14:57_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">n1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:57_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqfunc.html#D10" onmouseover="rs('seqfunc/D10')" onmouseout="rh()">SEQFUNC:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:57_1"/>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A13</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="c5">P</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> ) )
 
;<br/>
<a NAME="E10:57"/><span class="lab"><font color="Green" title="E49">A14</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/>( <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies ( <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</span><br/>

<a NAME="E2:57_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font> is   <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>;<br/>
<a NAME="E3:57_2"/><span class="kw">then </span>
(  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c7">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:57_2"/>
( <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c7">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A16</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
;<br/>
<a NAME="E12:57"/>
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="mesfun7c.html#D12" onmouseover="rs('mesfun7c/D12')" onmouseout="rh()">MESFUN7C:def 12</a></span>;<br/>
<a NAME="E13:57"/><span class="kw">then </span>
 <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E11:57"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="mesfun9c.html#T47" target="_self" onmouseover="rs('mesfun9c/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<a NAME="E14:57"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T25" onmouseover="rs('mesfun7c/T25')" onmouseout="rh()">MESFUN7C:25</a></span>;<br/>
<a NAME="E15:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A17</font></span>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<a NAME="E16:57"/><span class="lab"><font color="Green" title="E52">A18</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E49">A14</font></span></a></span>;<br/>
<a NAME="E17:57"/>
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="mesfun7c.html#D11" onmouseover="rs('mesfun7c/D11')" onmouseout="rh()">MESFUN7C:def 11</a></span>;<br/>
<a NAME="E18:57"/><span class="kw">then </span>
 <a href="mesfun7c.html#K8" title="MESFUN7C:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="txt" href="#E16:57"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="mesfun9c.html#T47" target="_self" onmouseover="rs('mesfun9c/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<a NAME="E19:57"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T25" onmouseover="rs('mesfun7c/T25')" onmouseout="rh()">MESFUN7C:25</a></span>;<br/>
<a NAME="E20:57"/><span class="kw">then </span>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E21:57"/>
 <a href="mesfun7c.html#K10" title="MESFUN7C:func.10">lim</a> <font color="Maroon" title="c6">F</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="mesfun6c.html#D2" onmouseover="rs('mesfun6c/D2')" onmouseout="rh()">MESFUN6C:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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