reserve i, j, n for Element of NAT,
  f, g, h, k for FinSequence of REAL,
  M, N for non empty MetrSpace;

theorem Th5:
  for f, g, h, k being FinSequence st f^g = h^k & len f = len h &
  len g = len k holds f = h & g = k
proof
  let f, g, h, k be FinSequence such that
A1: f^g = h^k and
A2: len f = len h and
A3: len g = len k;
A4: for i be Nat st 1 <= i & i <= len f holds f.i = h.i
  proof
    let i be Nat;
    assume
A5: 1 <= i & i <= len f;
    then
A6: i in dom h by A2,FINSEQ_3:25;
    i in dom f by A5,FINSEQ_3:25;
    hence f.i = (f^g).i by FINSEQ_1:def 7
      .= h.i by A1,A6,FINSEQ_1:def 7;
  end;
  for i be Nat st 1 <= i & i <= len g holds g.i = k.i
  proof
    let i be Nat;
    assume
A7: 1 <= i & i <= len g;
    then
A8: i in dom k by A3,FINSEQ_3:25;
    i in dom g by A7,FINSEQ_3:25;
    hence g.i = (f^g).(len f +i) by FINSEQ_1:def 7
      .= k.i by A1,A2,A8,FINSEQ_1:def 7;
  end;
  hence thesis by A2,A3,A4,FINSEQ_1:14;
end;
