reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th3:
  s1 = a GeoSeq iff s1.0 = 1 & for m holds s1.(m+1) = s1.m * a
proof
  defpred P[Nat] means s1.$1 = a GeoSeq.$1;
  hereby
    assume
A1: s1 = a GeoSeq;
    hence s1.0 = a|^0 by Def1
      .= 1 by NEWTON:4;
    let m;
    thus s1.(m+1) = a|^(m+1) by A1,Def1
      .= a|^m * a by NEWTON:6
      .= s1.m * a by A1,Def1;
  end;
  assume that
A2: s1.0 = 1 and
A3: for m holds s1.(m+1) = s1.m * a;
A4: for m st P[m] holds P[m+1]
  proof
    let m such that
A5: s1.m = a GeoSeq.m;
    thus s1.(m+1) = s1.m * a by A3
      .= a|^m * a by A5,Def1
      .= a|^(m+1) by NEWTON:6
      .= a GeoSeq.(m+1) by Def1;
  end;
  s1.0 = a|^0 by A2,NEWTON:4
    .= a GeoSeq.0 by Def1;
  then
A6: P[0];
  for m holds P[m] from NAT_1:sch 2(A6,A4);
  then for m being Element of NAT holds P[m];
  hence thesis by FUNCT_2:63;
end;
