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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Olive" title="b3">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp; <font color="Olive" title="b3">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">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">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> ) holds <br/> <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b2">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp; <font color="Olive" title="b2">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">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">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> ) holds <br/> <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font></span><br/><b>let </b><font color="Maroon" title="c3">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> &amp; <font color="Olive" title="b1">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp; <font color="Olive" title="b1">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">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">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="b2">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> ) holds <br/> <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</font></span><br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="mesfunc8.html#M1" title="MESFUNC8:mode.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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="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> <font color="Maroon" title="c3">E</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">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">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> ) implies  <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="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> <font color="Maroon" title="c3">E</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a>
 <b>and </b><br/><a NAME="E3:51"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>
 <b>and </b><br/><a NAME="E4:51"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">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">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</font>
 <b>and </b><br/><a NAME="E5:51"/><i><font color="Green" title="E47">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</font></span><br/>

<a NAME="E6:51"/><i><font color="Green" title="E48">P1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <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> <font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a>, <a class="txt" href="mesfunc9.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="mesfunc9.html#E2:51"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="mesfunc9.html#E3:51"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">PF</font> =  <a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c4">F</font> as  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="mesfunc9.html#E2:51"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="mesfunc9.html#E3:51"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="mesfunc9.html#T43" target="_self" title="MESFUNC9:th.43">ADD5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <font color="Maroon" title="c5">PF</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c6">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )</span><br/><b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E49">Q1</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">PF</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c6">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/><a NAME="E2:51_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c6">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>
 <b>by </b><i><a class="txt" href="mesfunc9.html#E5:51"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/><a NAME="E3:51_1"/><b>then </b>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c6">x</font></span>)</span> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc9.html#D2" target="_self" title="MESFUNC9:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E4:51_1"/>
<font color="Maroon" title="c5">PF</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c6">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="mesfunc9.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="mesfunc9.html#E2:51"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="mesfunc9.html#E3:51"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="mesfunc9.html#E1:51_1"><i><font color="Green" title="E49">Q1</font></i></a>, <a class="ref" href="mesfunc9.html#T33" target="_self" title="MESFUNC9:th.33">Cor01</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:51"/>
 <a href="mesfunc8.html#K8" title="MESFUNC8:func.8">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="mesfunc9.html#E6:51"><i><font color="Green" title="E48">P1</font></i></a>, <a class="txt" href="mesfunc9.html#E4:51"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mesfunc8.html#T25" title="MESFUNC8:th.25">MESFUNC8:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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