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

theorem thUnitalMagmaIsGroupCond: ::: GROUP_1
  for M being non empty unital multMagma
  st (for h being Element of M ex g being Element of M st h * g = 1_M
  & g * h = 1_M)
  holds M is Group-like
proof
  let M be non empty unital multMagma;
  assume A1: for h being Element of M
  ex g being Element of M
  st h * g = 1_M & g * h = 1_M;

  ex e being Element of M st
  for h being Element of M
  holds (h * e = h & e * h = h & ex g being Element of M st
  h * g = e & g * h = e)
  proof
    take e = 1_M;
    let h be Element of M;
    thus h * e = h by GROUP_1:def 4;
    thus e * h = h by GROUP_1:def 4;
    thus thesis by A1;
  end;
  hence M is Group-like;
end;
