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

theorem Th16:
  for z being Complex holds cos_C/.(<i>*z) = cosh_C/.z
proof
  let z be Complex;
  reconsider z as Element of COMPLEX by XCMPLX_0:def 2;
  cos_C/.(<i>*z) = (exp(<i>*<i>*z)+exp(-<i>*(<i>*z)))/2 by Def2
    .= (exp(-z)+exp(z))/2;
  then cos_C/.(<i>*z) = cosh_C/.z by Def4;
  hence thesis;
end;
