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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">X</font> being  <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a><br/> for <font color="Olive" title="b2">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b2">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> holds <br/>( <font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b2">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> &amp;  <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b2">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Olive" title="b2">seq</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</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">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> holds <br/>( <font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b1">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> &amp;  <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Olive" title="b1">seq</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">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="c3">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> implies ( <font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> &amp;  <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:10"/>
<font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4: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="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> &amp;  <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> )</span><br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green" title="E6">A1</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></i>;<br/>
<a NAME="E3:10"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</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">seq</font></span>)</span> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T5" title="BHSP_2:th.5">BHSP_2:5</a></i>;<br/>
<a NAME="E4:10"/><b>then </b>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span></span><br/>

<b>thus </b> <a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> = 
 <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</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">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" title="BHSP_4:func.2">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_4.html#E2:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="bhsp_2.html#T15" title="BHSP_2:th.15">BHSP_2:15</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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