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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></span>;<br/>
<a NAME="E3:10"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_2.html#T5" title="BHSP_2:th.5">BHSP_2:5</a></span>;<br/>
<a NAME="E4:10"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#T3" target="_self" title="BHSP_4:th.3">Th3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="bhsp_2.html#T15" title="BHSP_2:th.15">BHSP_2:15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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