
theorem
  for X being set, k being Nat st X c= Seg k holds rng Sgm X c= Seg k
proof
  let X be set, k be Nat;
  assume A1: X c= Seg k;
  for x being object holds x in rng Sgm X implies x in Seg k
  proof
    let x be object;
    assume A2: x in rng Sgm X;
    X is included_in_Seg by A1,FINSEQ_1:def 13;
    then rng Sgm X = X by FINSEQ_1:def 14;
    hence x in Seg k by A1, A2;
  end;
  hence thesis;
end;
