
theorem LemmaDiffConst:
  for f being Arithmetic_Progression,
      i being Nat holds
    f.(i+1) - f.i = f.1 - f.0
  proof
    let f be Arithmetic_Progression,
        i be Nat;
    dom f = NAT by FUNCT_2:def 1; then
    i in dom f & 0 in dom f & 1 in dom f & i+1 in dom f
      by ORDINAL1:def 12; then
    f.(i+1) - f.i = f.(0+1) - f.0 by APLikeDef;
    hence thesis;
  end;
