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

theorem
  b <> 0 & angle(a,b) = 0 implies angle(a,-b) = PI
proof
  assume that
A1: b <> 0 and
A2: angle(a,b) = 0;
A3: Arg(Rotate(b, -Arg a)) = 0 by A1,A2,Def3;
A4: Arg(Rotate(-b, -Arg a)) = Arg(-Rotate(b, -Arg a)) by Th56;
  Rotate(b, -Arg a) <> 0 by A1,Th50;
  then Arg -Rotate(b, -Arg a) = Arg Rotate(b, -Arg a)+PI by A3,Th12,COMPTRIG:5
    .= PI by A3;
  hence thesis by A1,A4,Def3;
end;
