reserve i for Integer,
  a, b, r, s for Real;

theorem
  for f being FinSequence, i being Nat st i+1 in dom f holds
  i in dom f or i = 0
proof
  let f be FinSequence;
  let i be Nat;
  assume
A1: i+1 in dom f;
  then 1 <= i+1 by FINSEQ_3:25;
  then
A2: 1 < i+1 or 1+0 = i+1 by XXREAL_0:1;
  per cases by A2,NAT_1:13;
  suppose
    i = 0;
    hence thesis;
  end;
  suppose
A3: 1 <= i;
    i+1 <= len f by A1,FINSEQ_3:25;
    then i <= len f by NAT_1:13;
    hence thesis by A3,FINSEQ_3:25;
  end;
end;
