reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th1:
  for z being Complex, e being Element of F_Complex st z = e
  holds n*z = n*e
proof
  let z be Complex,e be Element of F_Complex such that
A1: z=e;
  defpred P[Nat] means $1*z = $1 * e;
A2: P[0] by COMPLFLD:7,BINOM:12;
A3: for i being Nat st P[i] holds P[i+1]
    proof
      let i be Nat such that
A4:   P[i];
      (i+1)*e = e +(i*e) by BINOM:def 3
      .= z+(i*z) by A1,A4;
      hence thesis;
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
