
theorem
  for G1, G2 being Group for f being Homomorphism of G1, G2 for H2 being
  Subgroup of G2 ex H1 being strict Subgroup of G1 st the carrier of H1 = f"the
  carrier of H2
proof
  let G1, G2 be Group;
  let f be Homomorphism of G1, G2;
  let H2 be Subgroup of G2;
A1: for g being Element of G1 st g in f"the carrier of H2 holds g" in f"the
  carrier of H2
  proof
    let g be Element of G1;
    assume g in f"the carrier of H2;
    then f.g in the carrier of H2 by FUNCT_2:38;
    then f.g in H2 by STRUCT_0:def 5;
    then (f.g)" in H2 by GROUP_2:51;
    then f.g" in H2 by GROUP_6:32;
    then f.g" in the carrier of H2 by STRUCT_0:def 5;
    hence thesis by FUNCT_2:38;
  end;
A2: for g1, g2 being Element of G1 st g1 in f"the carrier of H2 & g2 in f"
  the carrier of H2 holds g1 * g2 in f"the carrier of H2
  proof
    let g1, g2 be Element of G1;
    assume that
A3: g1 in f"the carrier of H2 and
A4: g2 in f"the carrier of H2;
    f.g2 in the carrier of H2 by A4,FUNCT_2:38;
    then
A5: f.g2 in H2 by STRUCT_0:def 5;
    f.g1 in the carrier of H2 by A3,FUNCT_2:38;
    then f.g1 in H2 by STRUCT_0:def 5;
    then f.g1 * f.g2 in H2 by A5,GROUP_2:50;
    then f.(g1 * g2) in H2 by GROUP_6:def 6;
    then f.(g1 * g2) in the carrier of H2 by STRUCT_0:def 5;
    hence thesis by FUNCT_2:38;
  end;
  1_G2 in H2 by GROUP_2:46;
  then 1_G2 in the carrier of H2 by STRUCT_0:def 5;
  then f.1_G1 in the carrier of H2 by GROUP_6:31;
  then f"the carrier of H2 <> {} by FUNCT_2:38;
  then consider H1 being strict Subgroup of G1 such that
A6: the carrier of H1 = f"the carrier of H2 by A1,A2,GROUP_2:52;
  take H1;
  thus thesis by A6;
end;
