theorem
  for n being Nat for f being FinSequence holds 1 <= n & n+1 <= len
  f iff n in dom f & n+1 in dom f
proof
  let n be Nat;
  let f be FinSequence;
  thus 1<=n & n+1 <= len f implies n in dom f & n+1 in dom f
  proof
    assume that
A1: 1<=n and
A2: n+1 <= len f;
    n<=n+1 by NAT_1:11;
    then 1<=n+1 & n<=len f by A2,NAT_1:11,XXREAL_0:2;
    hence thesis by A1,A2,FINSEQ_3:25;
  end;
  thus thesis by FINSEQ_3:25;
end;
