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

theorem Th61:
  r > 0 implies angle(a,b) = angle(a*r,b*r)
proof
  assume
A1: r > 0;
  then
A2: Arg(a*r) = Arg a by Th25;
  per cases;
  suppose
A3: b <> 0;
    hence angle(a,b) = Arg(Rotate(b,-Arg a)) by Def3
      .= Arg(Rotate(b*r,-Arg(a*r))) by A1,A2,A3,Th25,Th60
      .= angle(a*r,b*r) by A1,A3,Def3;
  end;
  suppose
A4: b = 0;
    thus thesis
    proof
      per cases;
      suppose
A5:     Arg a = 0;
        hence angle(a,b) = Arg(Rotate(b,-Arg a)) by Def3
          .= Arg 0c by A4,Th50
          .= Arg(Rotate(b*r,-Arg(a*r))) by A4,Th50
          .= angle(a*r,b*r) by A2,A5,Def3;
      end;
      suppose
A6:     Arg a <> 0;
        hence angle(a,b) = 2*PI-Arg a by A4,Def3
          .= angle(a*r,b*r) by A2,A4,A6,Def3;
      end;
    end;
  end;
end;
