reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th60:
  for a,b being Complex holds Arg a = Arg b & a <> 0 & b <>
  0 implies Arg Rotate(a,r) = Arg Rotate(b,r)
proof
  let a,b be Complex;
  assume that
A1: Arg a = Arg b and
A2: a <> 0 and
A3: b <> 0;
  consider i being Integer such that
A4: Arg(Rotate(a,r)) = 2*PI*i+(r+Arg(a)) by A2,Th52;
  consider j being Integer such that
A5: Arg(Rotate(b,r)) = 2*PI*j+(r+Arg(b)) by A3,Th52;
A6: 0 <= Arg Rotate(a,r) & 0 <= Arg Rotate(b,r) by COMPTRIG:34;
A7: Arg Rotate(a,r) < 2*PI & Arg Rotate(b,r) < 2*PI by COMPTRIG:34;
  Arg(Rotate(b,r)) = 2*PI*(j-i)+Arg(Rotate(a,r)) by A1,A4,A5;
  hence thesis by A6,A7,Th2;
end;
