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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">seq1</font>, <font color="Olive" title="b2">seq2</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq1</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp; <font color="Olive" title="b2">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> holds <br/>( <font color="Olive" title="b1">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Olive" title="b2">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp;  <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Olive" title="b2">seq2</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</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="c2">seq1</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp; <font color="Maroon" title="c3">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> implies ( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp;  <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c2">seq1</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp; <font color="Maroon" title="c3">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp;  <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> )</span><br/>

<a NAME="E2:25"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp;  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> )
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="lopban_3.html#D2" target="_self" title="LOPBAN_3:def.2">Def2</a></i>;<br/>
<a NAME="E3:25"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green" title="E24">A4</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopban_3.html#T21" target="_self" title="LOPBAN_3:th.21">Th21</a></i>;<br/>
 <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font></span>)</span> = 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T21" target="_self" title="LOPBAN_3:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>
;<br/>
<b>hence </b><a NAME="E6:25"/>
( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp;  <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:25"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E4:25"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="lopban_3.html#D2" target="_self" title="LOPBAN_3:def.2">Def2</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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