reserve n for Nat;

theorem Th5:
  for D be set for f be FinSequence of D holds f is
weakly-one-to-one iff for i be Nat st 1 <= i & i < len f holds f/.i
  <> f/.(i+1)
proof
  let D be set;
  let f be FinSequence of D;
  thus f is weakly-one-to-one implies for i be Nat st 1 <= i & i <
  len f holds f/.i <> f/.(i+1)
  proof
    assume
A1: f is weakly-one-to-one;
    let i be Nat such that
A2: 1 <= i and
A3: i < len f;
A4: i+1 <= len f by A3,NAT_1:13;
    1 < i+1 by A2,NAT_1:13;
    then
A5: f.(i+1) = f/.(i+1) by A4,FINSEQ_4:15;
    f.i = f/.i by A2,A3,FINSEQ_4:15;
    hence thesis by A1,A2,A3,A5;
  end;
  assume
A6: for i be Nat st 1 <= i & i < len f holds f/.i <> f/.(i+1 );
  now
    let i be Nat such that
A7: 1 <= i and
A8: i < len f;
A9: i+1 <= len f by A8,NAT_1:13;
    1 < i+1 by A7,NAT_1:13;
    then
A10: f.(i+1) = f/.(i+1) by A9,FINSEQ_4:15;
    f.i = f/.i by A7,A8,FINSEQ_4:15;
    hence f.i <> f.(i+1) by A6,A7,A8,A10;
  end;
  hence thesis;
end;
