reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem
  for s being Real holds Rotate(s) = Rotate(s+2*PI*i)
  proof
    let s be Real;
    let p be Point of T2;
    set q = p`1+(p`2)*<i>;
A1: Rotate(q,s) = Rotate(q,s+2*PI*i) by Th39;
    thus (Rotate(s)).p = |[Re Rotate(q,s),Im Rotate(q,s)]| by JORDAN24:def 3
    .= (Rotate(s+2*PI*i)).p by A1,JORDAN24:def 3;
  end;
