reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem Th31:
  for x being 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+cos x*sin(n*x))*
    <i>
      .= cos(n*x+x)+(cos(n*x)*sin x+cos x*sin(n*x))*<i> by SIN_COS:75
      .= 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 be Nat holds P[n] from NAT_1:sch 2(A2,A1);
end;
