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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">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="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="b2">E</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="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">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="b2">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_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  ) holds <br/> 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="b2">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b4">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> </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">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">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">E</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="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">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="Olive" title="b1">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_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  ) holds <br/> 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="Olive" title="b1">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b3">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">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="Maroon" title="c3">E</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="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">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="c3">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_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  ) holds <br/> 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="c3">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b2">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</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="c3">E</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="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">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="c3">E</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  ) implies  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="c3">E</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c3">E</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="E2:31"/><span class="lab"><font color="Green" title="E30">A2</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="E3:31"/><span class="lab"><font color="Green" title="E31">A3</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="c3">E</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</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>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Olive" title="b1">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> </span><br/>

<a NAME="E4:31"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</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="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="mesfun9c.html#T2" target="_self" onmouseover="rs('mesfun9c/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E5:31"/><span class="lab"><font color="Green" title="E33">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>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" 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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <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="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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#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="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> </span><br/>

<a NAME="E2:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E3:31_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"><font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:31_1"/>
<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="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E34">A6</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="E6:31"/><span class="lab"><font color="Green" title="E34">A7</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="c3">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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" 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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <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="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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="c4">F</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> </span><br/>

<a NAME="E2:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E3:31_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="c4">F</font> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:31_2"/>
<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="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E35">A8</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>
<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"> verum</span>
<div class="add"><span class="kw">let </span><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" 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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">E</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:31_3"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> </span><br/><a NAME="E2:31_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#D1" target="_self" onmouseover="rs('mesfun9c/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:31_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</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="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/><a NAME="E4:31_3"/>
 <a href="mesfun7c.html#K12" title="MESFUN7C:func.12">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#T22" target="_self" onmouseover="rs('mesfun9c/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E5:31_3"/><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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="mesfun9c.html#T6" target="_self" onmouseover="rs('mesfun9c/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E6:31_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A11</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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="mesfun7c.html#T23" onmouseover="rs('mesfun7c/T23')" onmouseout="rh()">MESFUN7C:23</a></span>;<br/><a NAME="E7:31_3"/>
 <a href="mesfun7c.html#K11" title="MESFUN7C:func.11">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun9c.html#T21" target="_self" onmouseover="rs('mesfun9c/T21')" onmouseout="rh()">Th21</a></span>;<br/><a NAME="E8:31_3"/><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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c6">x</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="mesfun9c.html#T6" target="_self" onmouseover="rs('mesfun9c/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E9:31_3"/><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"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E36">A10</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="E10:31_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="mesfun9c.html#K1" title="MESFUN9C:func.1">||</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="mesfun7c.html#K9" title="MESFUN7C:func.9">#</a> <font color="Maroon" title="c6">x</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_3"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="comseq_3.html#T57" onmouseover="rs('comseq_3/T57')" onmouseout="rh()">COMSEQ_3:57</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>


</div>
