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

theorem
  r < 0 & a <> 0 & b <> 0 implies angle(a,b) = angle(a*r,b*r)
proof
  assume that
A1: r < 0 and
A2: a <> 0 and
A3: b <> 0;
  consider i being Integer such that
A4: Arg(Rotate(-b, -Arg -a)) = 2*PI*i+(-Arg -a + Arg -b) by A3,Th52;
  set br = b*r, ar = a*r;
  Arg(b*r) = Arg -b & Arg(a*r) = Arg -a by A1,Th26;
  then consider j being Integer such that
A5: Arg(Rotate(br,-Arg ar)) = 2*PI*j+(-Arg -a + Arg -b) by A1,A3,Th52;
A6: Arg(Rotate(br,-Arg ar)) = 2*PI*(j-i)+Arg(Rotate(-b,-Arg -a)) by A4,A5;
A7: 0 <= Arg(Rotate(br,-Arg ar)) & Arg(Rotate(br,-Arg ar)) < 2*PI by
COMPTRIG:34;
A8: 0 <= Arg(Rotate(-b,-Arg -a)) & Arg(Rotate(-b,-Arg -a)) < 2*PI by
COMPTRIG:34;
  thus angle(a,b) = angle(Rotate(a,PI),Rotate(b,PI)) by A2,A3,Th63
    .= angle(-a,Rotate(b,PI)) by Th58
    .= angle(-a,-b) by Th58
    .= Arg(Rotate(-b,-Arg -a)) by A3,Def3
    .= Arg(Rotate(br,-Arg ar)) by A6,A7,A8,Th2
    .= angle(a*r,b*r) by A1,A3,Def3;
end;
