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 Th49:
  h is Homomorphism of G,Image h
proof
  rng h = the carrier of Image h & dom h = the carrier of G by Th44,
FUNCT_2:def 1;
  then reconsider f9 = h as Function of G, Image h by RELSET_1:4;
  now
    let a,b;
    thus f9.a * f9.b = h.a * h.b by GROUP_2:43
      .= f9.(a * b) by Def6;
  end;
  hence thesis by Def6;
end;
