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

theorem Th14:
  cosh_C/.(z1+z2) = (cosh_C/.z1)*(cosh_C/.z2) + (sinh_C/.z1)*( sinh_C/.z2)
proof
  cosh_C/.(z1+z2) = cosh_C/.(z1- -z2)
    .=(cosh_C/.z1)*cosh_C/.(-z2) - (sinh_C/.z1)*sinh_C/.(-z2) by Th13
    .=(cosh_C/.z1)*cosh_C/.(z2) - (sinh_C/.z1)*sinh_C/.(-z2) by Th10
    .=(cosh_C/.z1)*cosh_C/.(z2) - (sinh_C/.z1)*(-sinh_C/.z2) by Th9
    .=(cosh_C/.z1)*cosh_C/.(z2) - -(sinh_C/.z1)*(sinh_C/.z2);
  hence thesis;
end;
