reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;

theorem Th11:
  (for G being trivial Group holds card G = 1 & G is finite) &
  for G being Group st card G = 1 holds G is trivial
proof
  thus for G being trivial Group holds card G = 1 & G is finite
  proof
    let G be trivial Group;
    ex x being object st the carrier of G = {x} by Def2;
    hence thesis by CARD_1:30;
  end;
  let G be Group;
  assume card G = 1;
  hence ex x being object st the carrier of G = {x} by CARD_2:42;
end;
