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

theorem Th54:
  Re Rotate(a,r) = (Re a)*(cos r)-(Im a)*(sin r) & Im Rotate(a,r)
  = (Re a)*(sin r)+(Im a)*(cos r)
proof
  a = |.a.|*cos Arg a+(|.a.|*sin Arg a)*<i> by COMPTRIG:62;
  then
A1: Re a = |.a.|*cos Arg a & Im a = |.a.|*sin Arg a by COMPLEX1:12;
  thus Re Rotate(a,r) = |.a.|*cos (r+Arg a) by COMPLEX1:12
    .= |.a.|*((cos r)*cos(Arg a)-(sin r)*sin(Arg a)) by SIN_COS:75
    .= (Re a)*(cos r)-(Im a)*(sin r) by A1;
  thus Im Rotate(a,r) = |.a.|*sin(r+Arg a) by COMPLEX1:12
    .= |.a.|*((sin r)*cos(Arg a)+(cos r)*sin(Arg a)) by SIN_COS:75
    .= (Re a)*(sin r)+(Im a)*(cos r) by A1;
end;
