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 Th73:
  for G being Group holds x in G` iff ex F being FinSequence of
the carrier of G,I st len F = len I & rng F c= commutators G & x = Product(F |^
  I)
proof
  let G be Group;
  thus x in G` implies ex F being FinSequence of the carrier of G,I st len F =
  len I & rng F c= commutators G & x = Product(F |^ I)
  proof
    assume
A1: x in G`;
    then x in G by GROUP_2:40;
    then reconsider a = x as Element of G by STRUCT_0:def 5;
    ex F being FinSequence of the carrier of G,I st len F = len I & rng F
    c= commutators G & a = Product(F |^ I) by A1,GROUP_4:28;
    hence thesis;
  end;
  given F being FinSequence of the carrier of G,I such that
A2: len F = len I & rng F c= commutators G & x = Product(F |^ I);
  thus thesis by A2,GROUP_4:28;
end;
