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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Cseq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b1">seq</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Olive" title="b1">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="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Olive" title="b2">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Olive" title="b1">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="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Cseq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Olive" title="b1">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Cseq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E26">A1</font></span>: <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T11" title="CLVECT_1:th.11">CLVECT_1:11</a></span>
;<br/>
<span class="lab"><font color="Green" title="E27">A2</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</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" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E28">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:59_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:59_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></span>;<br/><a NAME="E3:59_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>;<br/><span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<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" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D3" title="CLVECT_1:def.3">CLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="BHSP_4: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="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T1" title="VALUED_1:th.1">VALUED_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</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" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<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" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<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" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
;<br/><a NAME="E5:59_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4: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="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/><a NAME="E6:59_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4: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="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E7:59_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#D1" title="BHSP_4:def.1">BHSP_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_3.html#D9" target="_self" title="CLVECT_3:def.9">Def9</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>
<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" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
;<br/>
<a NAME="E6:59"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4: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="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<a NAME="E7:59"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"><font color="Maroon" title="c3">Cseq</font> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="bhsp_4.html#K1" title="BHSP_4: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="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">Cseq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 1</span>)</span></span>)</span> <a href="clvect_3.html#K6" title="CLVECT_3:func.6">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4: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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>;<br/>
<a NAME="E8:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:59"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E28">A3</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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