
theorem GROUP252INV:
  for G, H be Group, K be Subgroup of H, f be Homomorphism of G, H holds
  ex J be strict Subgroup of G
  st the carrier of J = f"(the carrier of K)
  proof
    let G, H be Group, K be Subgroup of H, f be Homomorphism of G,H;
    f.(1_G) = 1_H by GROUP_6:31;
    then f.(1_G) in K by GROUP_2:46;
    then reconsider
    Ivf = f"(the carrier of K) as non empty Subset of (the carrier of G)
    by FUNCT_2:38;
    D191: for g1, g2 being Element of G st g1 in Ivf & g2 in Ivf
    holds g1 * g2 in Ivf
    proof
      let g1, g2 be Element of G;
      D94: f.(g1*g2) = (f.g1)*(f.g2) by GROUP_6:def 6;
      assume g1 in Ivf & g2 in Ivf;
      then f.g1 in K & f.g2 in K by FUNCT_2:38;
      then (f.g1)*(f.g2) in K by GROUP_2:50;
      hence g1*g2 in Ivf by D94, FUNCT_2:38;
    end;
    for g being Element of G st g in Ivf holds g" in Ivf
    proof
      let g be Element of G;
      assume g in Ivf;
      then f.g in K by FUNCT_2:38;
      then (f.g)" in K by GROUP_2:51;
      then f.(g") in the carrier of K by GROUP_6:32;
      hence g" in Ivf by FUNCT_2:38;
    end;
    then
    consider J be strict Subgroup of G such that
    D19: the carrier of J = f"(the carrier of K) by GROUP_2:52, D191;
    take J;
    thus thesis by D19;
  end;
