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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  implies ( <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>  &amp;  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2"/>
<font color="Maroon" title="c1">seq</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>  &amp;  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> )</span><br/>

<a NAME="E2:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T16" title="MESFUNC9:th.16">MESFUNC9:16</a></span>;<br/>
<a NAME="E3:2"/><span class="kw">then </span>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:2"/>
<font color="Maroon" title="c1">seq</font> is  <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#D2" title="MESFUNC9:def.2">MESFUNC9:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c1">seq</font></span><br/>

<a NAME="E5:2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2"/>
 <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K19" title="SUPINF_2:func.19">SUM</a> <font color="Maroon" title="c1">seq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#T1" target="_self" title="MEASURE8:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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