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 Th57:
  for H being strict Group, h being Homomorphism of G,H holds
  h is onto iff Image h = H
proof
  let H be strict Group, h be Homomorphism of G,H;
  thus h is onto implies Image h = H
  proof
    assume rng h = the carrier of H;
    then the carrier of Image h = the carrier of H by Th44;
    hence thesis by GROUP_2:61;
  end;
  assume
A1: Image h = H;
  the carrier of H c= rng h
  proof
    let x be object;
    assume x in the carrier of H;
    then x in h .: (the carrier of G) by A1,Def10;
    then ex y being object st y in dom h & y in the carrier of G & h.y = x
      by FUNCT_1:def 6;
    hence thesis by FUNCT_1:def 3;
  end;
  then rng h = the carrier of H;
  hence thesis;
end;
