reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;

theorem
  for n holds (seq*Nseq).n = seq.(Nseq.n)
proof
  let n;
A1: n in NAT by ORDINAL1:def 12;
  dom (seq*Nseq) = NAT by FUNCT_2:def 1;
  hence thesis by FUNCT_1:12,A1;
end;
