reserve X for set;

theorem Th6:
  for G being Group
  holds G is trivial iff the multMagma of G = (1).G
proof
  let G be Group;
  thus G is trivial implies the multMagma of G = (1).G
  proof
    assume G is trivial;
    then consider x being object such that
    A1: the carrier of G = {x};
    x = 1_G by A1, TARSKI:def 1;
    then the carrier of G = the carrier of (1).G by A1,GROUP_2:def 7;
    hence the multMagma of G = (1).G by GROUP_2:61;
  end;
  thus the multMagma of G = (1).G implies G is trivial;
end;
