
theorem TAYLOR21:
  for a be Complex, n be Nat holds |.a.||^n = |.a|^n.|
  proof
    let a be Complex, n be Nat;
    defpred P[Nat] means |.a.||^$1 = |.a|^$1.|;
    A1: P[0]
    proof
      |.a.||^0 = |.1.| by NEWTON:4
      .= |.a|^0.| by NEWTON:4;
      hence thesis;
    end;
    A2: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
      B1: |.a.||^k = |.a|^k.|;
      |.a|^(k+1).| = |.a*a|^k.| by NEWTON:6
      .= |.a.|*|.a.| |^k by B1,COMPLEX1:65;
      hence thesis by NEWTON:6;
    end;
:: remark: the sentence (and similar):
:: for l be Nat holds  |.a.||^l = |.a|^l.| from NAT_1:sch 2(A1,A2);
:: works with relprem, but not with the verifier (resulting in 23 error).
    for l be Nat holds P[l] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
