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 Th61:
  x in [.A,B.] iff ex F,I st len F = len I & rng F c= commutators(
  A,B) & x = Product(F |^ I)
proof
  thus x in [.A,B.] implies ex F,I st len F = len I & rng F c= commutators(A,B
  ) & x = Product(F |^ I)
  proof
    assume
A1: x in [.A,B.];
    then x in G by GROUP_2:40;
    then reconsider a = x as Element of G by STRUCT_0:def 5;
    a in gr commutators(A,B) by A1;
    hence thesis by GROUP_4:28;
  end;
  thus thesis by GROUP_4:28;
end;
