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
  commutators({a},{b}) = {[.a,b.]}
proof
  thus commutators({a},{b}) c= {[.a,b.]}
  proof
    let x be object;
    assume x in commutators({a},{b});
    then consider c,d such that
A1: x = [.c,d.] and
A2: c in {a} & d in {b};
    c = a & b = d by A2,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x be object;
  assume x in {[.a,b.]};
  then
A3: x = [.a,b.] by TARSKI:def 1;
  a in {a} & b in {b} by TARSKI:def 1;
  hence thesis by A3;
end;
