
theorem
  for G1, G2 being Group for f being Homomorphism of G1, G2 for H1, H2
being Subgroup of G2 for H3, H4 being Subgroup of G1 st the carrier of H3 = f"
  the carrier of H1 & the carrier of H4 = f"the carrier of H2 holds H1 is
  Subgroup of H2 implies H3 is Subgroup of H4
proof
  let G1, G2 be Group;
  let f be Homomorphism of G1, G2;
  let H1, H2 be Subgroup of G2;
  let H3, H4 be Subgroup of G1 such that
A1: the carrier of H3 = f"the carrier of H1 & the carrier of H4 = f"the
  carrier of H2;
  assume H1 is Subgroup of H2;
  then the carrier of H1 c= the carrier of H2 by GROUP_2:def 5;
  hence thesis by A1,GROUP_2:57,RELAT_1:143;
end;
