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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/>



<a NAME="E1:39"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T37" target="_self" title="BHSP_4:th.37">Th37</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E2:39"/><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="seqm_3.html#T21" title="SEQM_3:th.21">SEQM_3:21</a></i>;<br/>
<a NAME="E3:39"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0</span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T34" title="BHSP_1:th.34">BHSP_1:34</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="bhsp_2.html#D3" title="BHSP_2:def.3">BHSP_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:39"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="bhsp_4.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/>


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