reserve x for Real;

theorem Th53:
  for x be Real, n be Nat holds (cos x+(sin x)*<i>)|^n =
  cos (n*x)+sin (n*x)*<i>
proof
  let x be Real;
  defpred P[Nat] means (cos x+(sin x)*<i>)|^$1 = cos ($1*x)+sin ($1*x)*<i>;
A1: now
    let n be Nat;
    assume P[n];
    then
    (cos x+(sin x)*<i>)|^(n+1) = (cos (n*x)+sin (n*x)*<i>)*(cos x+(sin x)*
    <i>) by NEWTON:6
      .= cos(n*x)*cos x - sin(n*x)*sin x +cos(n*x)*(sin x)*<i>+cos x*sin(n*x
    )*<i>
      .= cos(n*x+x)+cos(n*x)*(sin x)*<i>+cos x*sin(n*x)*<i> by SIN_COS:75
      .= cos(n*x+x)+(cos(n*x)*sin x+cos x*sin(n*x))*<i>
      .= cos ((n+1)*x)+sin ((n+1)*x)*<i> by SIN_COS:75;
    hence P[n+1];
  end;
A2: P[0] by NEWTON:4,SIN_COS:31;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A2,A1);
end;
