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

theorem Th37:
  for z being Complex holds exp(-<i>*z) = cos_C/.z - <i>* sin_C/.z
proof
  let z be Complex;
  reconsider z as Element of COMPLEX by XCMPLX_0:def 2;
  cos_C/.z - <i>*sin_C/.z = (exp(<i>*z) + exp(-<i>*z))/(2) - <i>*sin_C/.z
  by Def2
    .= (exp(<i>*z) + exp(-<i>*z))/2 - <i>*((exp(<i>*z) - exp(-<i>*z))/(2*<i>
  )) by Def1
    .= (exp(-<i>*z) + exp(-<i>*z))/2
    .= (Re exp(-<i>*z) + Re exp(-<i>*z)+(Im exp(-<i>*z) + Im exp(-<i>*z))*
  <i>)/2 by COMPLEX1:81
    .= (2*Re exp(-<i>*z)+2*Im exp(-<i>*z)*<i>)/2
    .= (Re(2*exp(-<i>*z))+2*Im exp(-<i>*z)*<i>)/2 by COMSEQ_3:17
    .= (Re(2*exp(-<i>*z))+Im(2*exp(-<i>*z))*<i>)/2 by COMSEQ_3:17
    .= (2*exp(-<i>*z))/2 by COMPLEX1:13;
  hence thesis;
end;
