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

theorem
  cos_C/.(z1-z2) = (cos_C/.z1)*(cos_C/.z2) + (sin_C/.z1)*(sin_C/.z2)
proof
  cos_C/.(z1-z2) = cos_C/.(z1+-z2)
    .=cos_C/.z1*cos_C/.(-z2) - sin_C/.z1*sin_C/.(-z2) by Th6
    .=cos_C/.z1*cos_C/.z2 - sin_C/.z1*sin_C/.(-z2) by Th3
    .=cos_C/.z1*cos_C/.z2 - sin_C/.z1*(-sin_C/.z2) by Th2
    .=cos_C/.z1*cos_C/.z2 - -sin_C/.z1*sin_C/.z2;
  hence thesis;
end;
