reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th55:
  G is commutative Group iff for A,B st B <> {} holds A |^ B = A
proof
  thus G is commutative Group implies for A,B st B <> {} holds A |^ B = A
  proof
    assume
A1: G is commutative Group;
    let A,B;
    set y = the Element of B;
    assume
A2: B <> {};
    then reconsider y as Element of G by TARSKI:def 3;
    thus A |^ B c= A
    proof
      let x be object;
      assume x in A |^ B;
      then ex a,b st x = a |^ b & a in A & b in B;
      hence thesis by A1,Th29;
    end;
    let x be object;
    assume
A3: x in A;
    then reconsider a = x as Element of G;
    a |^ y = x by A1,Th29;
    hence thesis by A2,A3;
  end;
  assume
A4: for A,B st B <> {} holds A |^ B = A;
  now
    let a,b;
    {a} = {a} |^ {b} by A4
      .= {a |^ b} by Th37;
    hence a = a |^ b by ZFMISC_1:3;
  end;
  hence thesis by Th30;
end;
