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 Th57:
  G is commutative Group iff for H1,H2 holds commutators(H1,H2) = {1_G}
proof
  thus G is commutative Group implies for H1,H2 holds commutators(H1,H2) = {
  1_G} by Th51;
  assume
A1: for H1,H2 holds commutators(H1,H2) = {1_G};
  G is commutative
  proof
    let a,b;
    b in {b} by TARSKI:def 1;
    then
A2: b in gr{b} by GROUP_4:29;
    a in {a} by TARSKI:def 1;
    then a in gr{a} by GROUP_4:29;
    then [.a,b.] in commutators(gr{a},gr{b}) by A2,Th52;
    then [.a,b.] in {1_G} by A1;
    then [.a,b.] = 1_G by TARSKI:def 1;
    hence thesis by Th36;
  end;
  hence thesis;
end;
