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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">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">D</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  &amp; <font color="Olive" title="b2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">D</font> holds <br/>(  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Olive" title="b3">F</font></span>)</span> <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>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">D</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b2">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  &amp; <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/>(  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be    <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">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#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b1">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  &amp; <font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</font> holds <br/>(  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</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#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">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  &amp; <font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</font> implies (  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> 
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D</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"> (  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</span><br/>

<a NAME="E4:39"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun7c.html#D11" onmouseover="rs('mesfun7c/D11')" onmouseout="rh()">MESFUN7C:def 11</a></span>;<br/>
<a NAME="E5:39"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="mesfun7c.html#D12" onmouseover="rs('mesfun7c/D12')" onmouseout="rh()">MESFUN7C:def 12</a></span>;<br/>
<a NAME="E6:39"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
(  <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="mesfun9c.html#K3" title="MESFUN9C:func.3">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#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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;  <a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="mesfun9c.html#T32" target="_self" onmouseover="rs('mesfun9c/T32')" onmouseout="rh()">Th32</a>, <a class="ref" href="mesfun9c.html#T34" target="_self" onmouseover="rs('mesfun9c/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c4">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/>
<a NAME="E8:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c4">F</font> is  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#T25" target="_self" onmouseover="rs('mesfun9c/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<div><span class="kw">hereby </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"> ( <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  implies  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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"> <span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> </span><br/><a NAME="E2:39_1"/><span class="kw">then </span>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 ;<br/><a NAME="E3:39_1"/><span class="kw">then </span>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T26" onmouseover="rs('comseq_3/T26')" onmouseout="rh()">COMSEQ_3:26</a></span>;<br/><a NAME="E4:39_1"/><span class="kw">then </span>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/><a NAME="E5:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="mesfun9c.html#T13" target="_self" onmouseover="rs('mesfun9c/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E6:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="ref" href="mesfun9c.html#T29" target="_self" onmouseover="rs('mesfun9c/T29')" onmouseout="rh()">Th29</a></span>;<br/><a NAME="E7:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/><a NAME="E8:39_1"/>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> 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="#E1:39_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/><a NAME="E9:39_1"/><span class="kw">then </span>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T26" onmouseover="rs('comseq_3/T26')" onmouseout="rh()">COMSEQ_3:26</a></span>;<br/><a NAME="E10:39_1"/><span class="kw">then </span>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/><a NAME="E11:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="mesfun9c.html#T13" target="_self" onmouseover="rs('mesfun9c/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E12:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="ref" href="mesfun9c.html#T29" target="_self" onmouseover="rs('mesfun9c/T29')" onmouseout="rh()">Th29</a></span>;<br/><a NAME="E13:39_1"/><span class="kw">then </span>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/><span class="kw">hence </span><a NAME="E14:39_1"/>
<span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">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="#E7:39_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="comseq_3.html#T42" onmouseover="rs('comseq_3/T42')" onmouseout="rh()">COMSEQ_3:42</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>

<span class="kw">assume </span><a NAME="E10:39"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">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="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> </span><br/>

<a NAME="E11:39"/><span class="kw">then </span>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
;<br/>
<a NAME="E12:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E13:39"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E14:39"/>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E10:39"><span class="lab"><font color="Green" title="E42">A11</font></span></a></span>;<br/>
<a NAME="E15:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <span class="p2">(<span class="default"><a href="mesfun9c.html#K3" title="MESFUN9C:func.3">Partial_Sums</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E16:39"/>
(  <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="mesfun9c.html#T13" target="_self" onmouseover="rs('mesfun9c/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E17:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> 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="#E12:39"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="mesfun9c.html#T29" target="_self" onmouseover="rs('mesfun9c/T29')" onmouseout="rh()">Th29</a>, <a class="ref" href="comseq_3.html#T26" onmouseover="rs('comseq_3/T26')" onmouseout="rh()">COMSEQ_3:26</a></span>;<br/>
<a NAME="E18:39"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<span class="p1">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/>
<a NAME="E19:39"/>
(  <a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">x</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  iff <span class="p1">(<span class="default"><a href="mesfun9c.html#K2" title="MESFUN9C:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c2">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="#E3:39"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="mesfun9c.html#T13" target="_self" onmouseover="rs('mesfun9c/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E20:39"/><span class="kw">then </span>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> 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="#E15:39"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="mesfun9c.html#T29" target="_self" onmouseover="rs('mesfun9c/T29')" onmouseout="rh()">Th29</a>, <a class="ref" href="comseq_3.html#T26" onmouseover="rs('comseq_3/T26')" onmouseout="rh()">COMSEQ_3:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:39"/>
 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c2">x</font></span>)</span> 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="#E17:39"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="comseq_3.html#T42" onmouseover="rs('comseq_3/T42')" onmouseout="rh()">COMSEQ_3:42</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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