
theorem ArDefNth:
  for a,r being Real,
      i being Nat holds
    ArProg (a,r).i = a + i * r
  proof
    let a,r be Real,
        i be Nat;
    defpred P[Nat] means ArProg (a,r).$1 = a + $1 * r;
    ArProg (a,r).0 = a + 0 * r by ArDefRec; then
A1: P[0];
A2: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A3:   P[k];
      ArProg (a,r).(k+1) = ArProg (a,r).k + r by ArDefRec;
      hence thesis by A3;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
