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;
