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 Th58:
  x in commutators G iff ex a,b st x = [.a,b.]
proof
  thus x in commutators G implies ex a,b st x = [.a,b.]
  proof
    assume x in commutators G;
    then ex a,b st x = [.a,b.] & a in (Omega).G & b in (Omega).G by Th52;
    hence thesis;
  end;
  given a,b such that
A1: x = [.a,b.];
  thus thesis by A1;
end;
