reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem
  h is one-to-one implies h" is Homomorphism of Image h,G
proof
  assume
A1: h is one-to-one;
  reconsider Imh = Image h as Group;
  h is Function of G,Imh & rng h = the carrier of Imh by Th44,Th49;
  then reconsider h9 = h" as Function of Imh, G by A1,FUNCT_2:25;
  now
    let a,b be Element of Imh;
    reconsider a9 = a, b9 = b as Element of H by GROUP_2:42;
    a9 in Imh;
    then consider a1 being Element of G such that
A2: h.a1 = a9 by Th45;
    b9 in Imh;
    then consider b1 being Element of G such that
A3: h.b1 = b9 by Th45;
    thus h9.(a * b) = h9.(h.a1 * h.b1) by A2,A3,GROUP_2:43
      .= h9.(h.(a1 * b1)) by Def6
      .= a1 * b1 by A1,FUNCT_2:26
      .= h9.a * b1 by A1,A2,FUNCT_2:26
      .= h9.a * h9.b by A1,A3,FUNCT_2:26;
  end;
  hence thesis by Def6;
end;
