reserve i for Element of NAT;

theorem Th12:
  for G,H being Group, h being Homomorphism of G,H
  for A,B being Subgroup of G holds
  A is Subgroup of B implies h.:A is Subgroup of h.:B
proof
  let G,H be Group;
  let h be Homomorphism of G,H;
  let A,B be Subgroup of G;
  assume A is Subgroup of B;
  then the carrier of A c= the carrier of B by GROUP_2:def 5;
  then
A1: h.:(the carrier of A)c= h.:(the carrier of B ) by RELAT_1:123;
  the carrier of h.:B= h.:(the carrier of B) by Th8;
  then the carrier of h.:A c= the carrier of h.:B by A1,Th8;
  hence thesis by GROUP_2:57;
end;
