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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">Cseq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c4">n</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/>







<a NAME="E1:62"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_3.html#T52" target="_self" title="CLVECT_3:th.52">Th52</a></i>;<br/>
<a NAME="E2:62"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>;<br/>
<a NAME="E3:62"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<a NAME="E4:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D10" title="COMSEQ_1:def.10">COMSEQ_1:def 10</a></i>;<br/>
<a NAME="E6:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T26" title="COMSEQ_1:th.26">COMSEQ_1:26</a></i>;<br/>
<a NAME="E7:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T14" title="COMSEQ_1:th.14">COMSEQ_1:14</a></i>;<br/>
<a NAME="E8:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span></span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <font color="Maroon" title="c3">Cseq</font></span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T14" title="COMSEQ_1:th.14">COMSEQ_1:14</a></i>;<br/>
<a NAME="E9:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T21" title="COMSEQ_1:th.21">COMSEQ_1:21</a></i>;<br/>
<a NAME="E10:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K9" title="CLVECT_1:func.9">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p5">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_3.html#T46" target="_self" title="CLVECT_3:th.46">Th46</a></i>;<br/>
<a NAME="E11:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K9" title="CLVECT_1:func.9">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p5">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_3.html#T3" target="_self" title="CLVECT_3:th.3">Th3</a></i>;<br/>
<a NAME="E12:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p5">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D14" title="CLVECT_1:def.14">CLVECT_1:def 14</a></i>;<br/>
<a NAME="E13:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p5">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T4" title="CLVECT_1:th.4">CLVECT_1:4</a></i>;<br/>
<b>hence </b><a NAME="E14:62"/>
<span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p3">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 1</span>)</span> <a href="comseq_1.html#K6" title="COMSEQ_1:func.6">-</a> <font color="Maroon" title="c3">Cseq</font></span>)</span> <a href="clvect_3.html#K7" title="CLVECT_3:func.7">*</a> <span class="p4">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 ;<br/>


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