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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49">RLVECT_1:49</a></i>
;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E5:55"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/>
<a NAME="E7:55"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:55_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>;<br/><a NAME="E3:55_5"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K9">+</a> <span class="p0">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p0">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E5:55_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E6:55_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Rseq</font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Rseq</font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">Rseq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E7:55_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E9:55"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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