theorem
  G is commutative Group iff for a,b holds [.a,b.] = 1_G
proof
  thus G is commutative Group implies for a,b holds [.a,b.] = 1_G
  proof
    assume
A1: G is commutative Group;
    let a,b;
    a * b = b * a by A1,Lm1;
    hence thesis by Th36;
  end;
  assume
A2: for a,b holds [.a,b.] = 1_G;
  G is commutative
  proof
    let a,b;
    [.a,b.] = 1_G by A2;
    hence thesis by Th36;
  end;
  hence thesis;
end;
