<?xml version="1.0"?>
<div class="add">

<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">F</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st (  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> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font> ) holds <br/> for <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st (  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> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font> ) holds <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st (  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> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) holds <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) implies  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E22">A1</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="c4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">m</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"> <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E2:21"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="c3">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E1:21_1"/>
<font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>;<br/>
<a NAME="E2:21_1"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:21"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T45" onmouseover="rs('mesfunc9/T45')" onmouseout="rh()">MESFUNC9:45</a></span>;<br/>
<a NAME="E4:21"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K1" title="MESFUN7C:func.1">R_EAL</a> <span class="p2">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#T7" target="_self" onmouseover="rs('mesfun9c/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E5:21"/><span class="kw">then </span>
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">M</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:21"/>
<span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c5">m</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a> <font color="Maroon" title="c3">M</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"> verum</span><br/>


</div>
