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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> $1 is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>;<br/>
<b>assume </b><a NAME="E1:15"/>
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T35" title="BHSP_3:th.35">BHSP_3:35</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>
 ;<br/>

<a NAME="E2:15_1"/><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="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</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#D2" target="_self" title="BHSP_4:def.2">Def2</a></i>;<br/>
<a NAME="E3:15_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T44" title="BHSP_3:th.44">BHSP_3:44</a></i>;<br/>
<a NAME="E4:15_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T37" title="BHSP_3:th.37">BHSP_3:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">seq1</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E6:15_1"/><i><font color="Green" title="E14">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:15_1"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c4">seq1</font> 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#T1" title="BHSP_2:th.1">BHSP_2:1</a></i>;<br/>
<a NAME="E8:15_1"/>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c4">seq1</font>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E6:15_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="bhsp_4.html#T12" target="_self" title="BHSP_4:th.12">Th12</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E3:15_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="bhsp_4.html#E7:15_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="bhsp_2.html#T4" title="BHSP_2:th.4">BHSP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E10:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="bhsp_4.html#E4:15_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:15"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bhsp_4.html#E2:15"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="bhsp_4.html#E3:15"><i><font color="Green" title="E11">A2</font></i></a>);</i><br/>


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