
theorem LM204D:
  for G being Group, H, K be normal Subgroup of G
  st (the carrier of H) /\ (the carrier of K) = {1_G}
  holds (nat_hom H) | (the carrier of K) is one-to-one
  proof
    let G be Group, H, K be normal Subgroup of G;
    assume
    AS1: (the carrier of H) /\ (the carrier of K) = {1_G};
    set f = nat_hom H;
    set g = f| (the carrier of K);
    for x1, x2 be object st x1 in dom g & x2 in dom g & g.x1 = g.x2
    holds x1 = x2
    proof
      let x1, x2 be object;
      assume
      AS2: x1 in dom g & x2 in dom g & g.x1 = g.x2;
      then
      A1: x1 in (the carrier of K) & x1 in dom f by RELAT_1:57;
      reconsider y1= x1 as Element of G by AS2;
      A2: x2 in (the carrier of K) & x2 in dom f by AS2, RELAT_1:57;
      reconsider y2 = x2 as Element of G by AS2;
      A3: y1 * H = f.y1 by GROUP_6:def 8
      .= g.x1 by A1, FUNCT_1:49
      .= f.y2 by AS2, A2, FUNCT_1:49
      .= y2*H by GROUP_6:def 8;
      y1*(1_G) in y1*H by GROUP_2:46, GROUP_2:103;
      then y1 in y2*H by A3, GROUP_1:def 4;
      then consider h be Element of G such that
      A4: y1 = y2*h & h in H by GROUP_2:103;
      y1 in K & y2 in K by AS2, RELAT_1:57;
      then y1 in K & y2" in K by GROUP_2:51;
      then
      A6: y2"*y1 in K by GROUP_2:50;
      y2"*y1 in the carrier of H by A4, GROUP_1:13;
      then y2"*y1 in {1_G} by AS1, XBOOLE_0:def 4, A6;
      then y2"*y1 = 1_G by TARSKI:def 1;
      then y2" = y1" by GROUP_1:12;
      hence thesis by GROUP_1:9;
    end;
    hence thesis by FUNCT_1:def 4;
  end;
