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 Th63:
  for G being set, H being non empty set for h being Function of G,H holds
  for g1 being Function of H,G holds
  h is bijective & g1 = h" implies g1 is bijective
proof
  let G be set,H be non empty set;
  let h be Function of G,H, g1 be Function of H,G;
  assume
A1: h is bijective & g1 = h";
  then dom h = G & rng g1 = dom h by FUNCT_1:33,FUNCT_2:def 1;
  hence thesis by A1,FUNCT_2:def 3;
end;
