
theorem
  for G1, G2 being Group for f being Homomorphism of G1, G2 for H1 being
Subgroup of G1 holds ex H2 being strict Subgroup of G2 st the carrier of H2 = f
  .:the carrier of H1
proof
  let G1, G2 be Group;
  let f be Homomorphism of G1, G2;
  let H1 be Subgroup of G1;
  reconsider y = f.1_G1 as Element of G2;
A1: for g being Element of G2 st g in f.:the carrier of H1 holds g" in f.:
  the carrier of H1
  proof
    let g be Element of G2;
    assume g in f.:the carrier of H1;
    then consider x being Element of G1 such that
A2: x in the carrier of H1 and
A3: g = f.x by FUNCT_2:65;
    x in H1 by A2,STRUCT_0:def 5;
    then x" in H1 by GROUP_2:51;
    then
A4: x" in the carrier of H1 by STRUCT_0:def 5;
    f.x" = (f.x)" by GROUP_6:32;
    hence thesis by A3,A4,FUNCT_2:35;
  end;
A5: for g1, g2 being Element of G2 st g1 in f.:the carrier of H1 & g2 in f
  .:the carrier of H1 holds g1 * g2 in f.:the carrier of H1
  proof
    let g1, g2 be Element of G2;
    assume that
A6: g1 in f.:the carrier of H1 and
A7: g2 in f.:the carrier of H1;
    consider x being Element of G1 such that
A8: x in the carrier of H1 and
A9: g1 = f.x by A6,FUNCT_2:65;
    consider y being Element of G1 such that
A10: y in the carrier of H1 and
A11: g2 = f.y by A7,FUNCT_2:65;
A12: y in H1 by A10,STRUCT_0:def 5;
    x in H1 by A8,STRUCT_0:def 5;
    then x * y in H1 by A12,GROUP_2:50;
    then
A13: x * y in the carrier of H1 by STRUCT_0:def 5;
    f.(x * y) = f.x * f.y by GROUP_6:def 6;
    hence thesis by A9,A11,A13,FUNCT_2:35;
  end;
  1_G1 in H1 by GROUP_2:46;
  then dom f = the carrier of G1 & 1_G1 in the carrier of H1 by FUNCT_2:def 1
,STRUCT_0:def 5;
  then y in f.:the carrier of H1 by FUNCT_1:def 6;
  then consider H2 being strict Subgroup of G2 such that
A14: the carrier of H2 = f.:the carrier of H1 by A1,A5,GROUP_2:52;
  take H2;
  thus thesis by A14;
end;
