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
  seq is constant iff for n,m holds seq.n = seq.m
proof
  thus seq is constant implies for n,m holds seq.n = seq.m by VALUED_0:23;
  assume that
A1: for n,m holds seq.n = seq.m and
A2: seq is non constant;
  now
    let n be Nat;
    consider n1 be Nat such that
A3: seq.n1 <> seq.n by A2,VALUED_0:def 18;
    thus contradiction by A1,A3;
  end;
  hence thesis;
end;
