reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th64:
  |.z.| < 1 implies z GeoSeq is summable & Sum(z GeoSeq) = 1r/(1r- z)
proof
  set seq2 = NAT --> 1r;
  deffunc f(Nat) = z |^ ($1+1);
  consider seq1 such that
A1: for n holds seq1.n=f(n) from COMSEQ_1:sch 1;
  assume
A2: |.z.|<1;
  then
A3: lim(seq1)=0c by A1,Th44;
A4: now
    let n be Element of NAT;
    thus Partial_Sums(z GeoSeq).n = (1r - z |^ (n+1))/(1r-z) by A2,Th36,
COMPLEX1:48
      .= (1r - seq1.n)/(1r-z) by A1
      .= 1r * (seq2.n-seq1.n)/(1r-z) by COMPLEX1:def 4,FUNCOP_1:7
      .= (1r/(1r-z)) * (seq2.n+-seq1.n) by XCMPLX_1:74
      .= (1r/(1r-z)) * (seq2.n+(-seq1).n) by VALUED_1:8
      .= (1r/(1r-z)) * (seq2-seq1).n by VALUED_1:1
      .= ((1r/(1r-z))(#)(seq2-seq1)).n by VALUED_1:6;
  end;
A5: for n holds seq2.n=1r
    by ORDINAL1:def 12,FUNCOP_1:7;
  then
A6: seq2 is convergent by COMSEQ_2:9;
A7: seq1 is convergent by A2,A1,Th44;
  then
A8: seq2-seq1 is convergent by A6;
  hence Partial_Sums(z GeoSeq) is convergent by A4,FUNCT_2:63;
  lim(seq2-seq1)=lim(seq2)-lim(seq1) by A7,A6,COMSEQ_2:26
    .=1r by A3,A5,COMSEQ_2:10;
  then lim((1r/(1r-z))(#)(seq2-seq1))=1r/(1r-z) * 1r by A8,COMSEQ_2:18
    .=1r/(1r-z) by COMPLEX1:def 4;
  hence thesis by A4,FUNCT_2:63;
end;
