 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem Th68:
  G2 is trivial implies
  for phi being Homomorphism of G1,G2
  holds phi = 1:(G1,G2)
proof
  assume A1: G2 is trivial;
  let phi be Homomorphism of G1,G2;
  the multMagma of G2 = (1).G2 by A1,GROUP_22:6;
  then A2: the carrier of G2 = {1_G2} by GROUP_2:def 7;
  for g being Element of G1
  holds phi.g = (1:(G1,G2)).g
  proof
    let g be Element of G1;
    (G1 --> (1_G2)).g = 1_G2;
    then A3: (1:(G1,G2)).g = 1_G2 by GROUP_6:def 7;
    thus phi.g = 1_G2 by A2, TARSKI:def 1
               .= (1:(G1,G2)).g by A3;
  end;

  hence thesis by FUNCT_2:def 8;
end;
