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

theorem Th36:
  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) +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;
