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

theorem Th58:
  Rotate(a, PI) = -a
proof
A1: a = |.a.|*cos Arg a+(|.a.|*sin Arg a)*<i> by COMPTRIG:62;
A2: |.a.|*-sin Arg a = -(|.a.|*sin Arg a) .= - Im a by A1,COMPLEX1:12;
A3: |.a.|*-cos Arg a = -(|.a.|*cos Arg a) .= - Re a by A1,COMPLEX1:12;
  thus Rotate(a, PI) = |.a.|*-cos Arg a + |.a.|*sin (PI+Arg a) *<i> by
SIN_COS:79
    .= |.a.|*-cos Arg a + (|.a.|*-sin Arg a) *<i> by SIN_COS:79
    .= -a by A3,A2,COMPLEX1:83;
end;
