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

theorem
  exp(z) = cosh_C/.z + sinh_C/.z
proof
  cosh_C/.z + sinh_C/.z = (exp(z)+exp(-z))/2 + sinh_C/.z by Def4
    .= (exp(z)+exp(-z))/2 + (exp(z)-exp(-z))/2 by Def3
    .= (exp(z)+(exp(-z) + exp(z)-exp(-z)))/2
    .=(Re(exp(z))+Re(exp(z))+(Im(exp(z))+Im(exp(z)))*<i>)/2 by COMPLEX1:81
    .=((2*Re(exp(z))+2*Im(exp(z))*<i>))/2
    .=((Re(2*exp(z))+2*Im(exp(z))*<i>))/2 by COMSEQ_3:17
    .=((Re(2*exp(z))+Im(2*exp(z))*<i>))/2 by COMSEQ_3:17
    .=(2*exp(z))/2 by COMPLEX1:13
    .=exp(z)*1;
  hence thesis;
end;
