reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem
  sin_C/.(z1-z2) = (sin_C/.z1)*(cos_C/.z2) - (cos_C/.z1)*(sin_C/.z2)
proof
  sin_C/.(z1-z2) = sin_C/.(z1 + -z2)
    .=(sin_C/.z1)*(cos_C/.(-z2)) + (cos_C/.z1)*(sin_C/.(-z2)) by Th4
    .=(sin_C/.z1)*(cos_C/.z2) + (cos_C/.z1)*(sin_C/.(-z2)) by Th3
    .=(sin_C/.z1)*(cos_C/.z2) + (cos_C/.z1)*(-sin_C/.z2) by Th2
    .=(sin_C/.z1)*(cos_C/.z2) + -(cos_C/.z1)*(sin_C/.z2);
  hence thesis;
end;
