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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K6">1r</a>  &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">seq</font> <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"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>thus </b><a NAME="E2:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_3.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:49"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:49"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="comseq_3.html#E1"><i><font color="Green">Lemma33</font></i></a>);<br/></i>
<b>then </b><i><font color="Green">E45</font></i>:  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font> = 
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T2" target="_self">Th2</a>, <a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>


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