
theorem APAsArProg:
  for f being Arithmetic_Progression holds
    f = ArProg (f.0,difference f)
  proof
    let f be Arithmetic_Progression;
    set a = f.0;
    set r = f.1 - f.0;
    defpred P[Nat] means f.$1 = ArProg(a,r).$1;
A2: P[0] by ArDefRec;
A3: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
a5:   P[k];
      f.(k+1) - f.k = r by LemmaDiffConst; then
      f.(k+1) = f.k + r;
      hence thesis by a5,ArDefRec;
    end;
S1: for n being Nat holds P[n] from NAT_1:sch 2(A2,A3);
    for n being Element of NAT holds f.n = ArProg(a,r).n by S1;
    hence thesis by FUNCT_2:def 8;
  end;
