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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E2:2_1"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E1:2_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complex1.html#T118">COMPLEX1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E2:2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="csspace2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="csspace2.html#E2:2"><i><font color="Green">E4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:2"/>
 <a href="comseq_3.html#K7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_2.html#K1">*'</a> 
 <b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>


</div>
