 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem
  for G1,G2 being Group
  for H being Subgroup of G2
  for f1 being Homomorphism of G1,G2
  for f2 being Homomorphism of G1,H
  st f1 = f2
  holds Image f1 = Image f2
proof
  let G1,G2 be Group;
  let H be Subgroup of G2;
  let f1 be Homomorphism of G1,G2;
  let f2 be Homomorphism of G1,H;
  assume A1: f1 = f2;
  A2: Image f2 is strict Subgroup of G2 by GROUP_2:56;
  for g being Element of G2 holds g in Image f1 iff g in Image f2
  proof
    let g be Element of G2;
    hereby
      assume g in Image f1;
      then ex a being Element of G1 st g = f1.a by GROUP_6:45;
      hence g in Image f2 by A1, GROUP_6:45;
    end;
    assume g in Image f2;
    then ex a being Element of G1 st g = f2.a by GROUP_6:45;
    hence g in Image f1 by A1, GROUP_6:45;
  end;
  hence Image f1 = Image f2 by A2, GROUP_2:def 6;
end;
