reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  for G being strict Group holds G is commutative Group iff G` = (1).G
proof
  let G be strict Group;
  thus G is commutative Group implies G` = (1).G
  proof
    assume
A1: G is commutative Group;
    now
      let a be Element of G;
      assume a in G`;
      then a in gr {1_G} by A1,Th51;
      then a in gr({1_G} \ {1_G}) by GROUP_4:35;
      then a in gr {} the carrier of G by XBOOLE_1:37;
      hence a in (1).G by GROUP_4:30;
    end;
    then
A2: G` is Subgroup of (1).G by GROUP_2:58;
    (1).G is Subgroup of G` by GROUP_2:65;
    hence thesis by A2,GROUP_2:55;
  end;
  assume
A3: G` = (1).G;
  G is commutative
  proof
    let a,b be Element of G;
    [.a,b.] in G` by Th74;
    then [.a,b.] = 1_G by A3,Th1;
    hence thesis by Th36;
  end;
  hence thesis;
end;
