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

theorem
  for z being Element of COMPLEX holds (sin_C/.z)*(sin_C/.z) + (cos_C/.z
  )*(cos_C/.z) = 1
proof
  let z be Element of COMPLEX;
  set z1 = exp(<i>*z), z2 = exp(-<i>*z);
  (sin_C/.z)*(sin_C/.z) + (cos_C/.z)*(cos_C/.z) =(sin_C/.z)*(sin_C/.z) + (
  cos_C/.z)*((exp(<i>*z) + exp(-<i>*z))/2) by Def2
    .=(sin_C/.z)*(sin_C/.z) + ((z1 + z2)/2)*((z1 + z2)/2) by Def2
    .=(z1 - z2)/(2*<i>)*(sin_C/.z) + ((z1 + z2)/2)*((z1 + z2)/2) by Def1
    .=((z1 - z2)/(2*<i>))*((z1 - z2)/(2*<i>)) + ((z1 + z2)/2)*((z1 + z2)/2)
  by Def1
    .=((z1*z2 + z1*z2) + (z1*z2 + z1*z2))/4
    .=((1 + z1*exp(-(<i>*z))) + (z1*exp(-(<i>*z)) + z1*exp(-(<i>*z)))) /4 by
Lm3
    .=((1 + 1) + (z1*exp(-(<i>*z)) + z1*exp(-(<i>*z)))) /4 by Lm3
    .=((1 + 1) + (z1*exp(-(<i>*z)) + 1))/4 by Lm3
    .=(2 + 2)/4 by Lm3;
  hence thesis;
end;
