 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 Th78:
  for G being Group
  for H being Subgroup of G
  holds H is trivial iff (for x being Element of G holds x in H iff x = 1_G)
proof
  let G be Group;
  let H be Subgroup of G;
  thus H is trivial implies (for x being Element of G holds x in H iff x = 1_G)
  proof
    assume H is trivial;
    then consider h being object such that
    A1: the carrier of H = {h} by GROUP_6:def 2;
    1_H in {h} by A1;
    then 1_H = h by TARSKI:def 1;
    then A2: the carrier of H = {1_H} by A1;
    let x be Element of G;
    hereby
      assume x in H;
      then x in {1_H} by A2;
      then x = 1_H by TARSKI:def 1;
      hence x = 1_G by GROUP_2:44;
    end;
    assume x = 1_G;
    then x = 1_H by GROUP_2:44;
    hence x in H;
  end;
  assume A3: for x being Element of G holds x in H iff x = 1_G;
  for x being object holds x in the carrier of H iff x = 1_G
  proof
    let x be object;
    hereby
      assume x in the carrier of H;
      then A4: x in H;
      then x in G by GROUP_2:40;
      hence x = 1_G by A3, A4;
    end;
    assume x = 1_G;
    then x in H by A3;
    hence x in the carrier of H;
  end;

  then the carrier of H = {1_G} by TARSKI:def 1;
  hence H is trivial;
end;
