reserve i for Element of NAT;

theorem Th9:
  for G,H being Group, h being Homomorphism of G,H holds
  for A being strict Subgroup of G holds
  Image(h|A) is strict Subgroup of Image h
proof
  let G,H be Group;
  let h be Homomorphism of G,H;
  let A be strict Subgroup of G;
A1: the carrier of A c= the carrier of G & h.:(the carrier of G) = the
  carrier of Image h by GROUP_2:def 5,GROUP_6:def 10;
  the carrier of Image (h|A)=rng (h|A) by GROUP_6:44
    .=h.:(the carrier of A) by RELAT_1:115;
  hence thesis by A1,GROUP_2:57,RELAT_1:123;
end;
