 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 Th77:
  for G being Group
  holds G is trivial iff (for x being Element of G holds x = 1_G)
proof
  let G be Group;
  thus G is trivial implies for x being Element of G holds x = 1_G;
  assume A2: for x being Element of G holds x = 1_G;
  for x being object holds x in the carrier of G iff x in {1_G}
  proof
    let x be object;
    hereby
      assume x in the carrier of G;
      then x = 1_G by A2;
      hence x in {1_G} by TARSKI:def 1;
    end;
    assume x in {1_G};
    hence x in the carrier of G;
  end;
  hence G is trivial by TARSKI:2;
end;
