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 Th38:
  a in H & b in H implies [.a,b.] in H
proof
  assume
A1: a in H & b in H;
  then a" in H & b" in H by GROUP_2:51;
  then
A2: a" * b" in H by GROUP_2:50;
  a * b in H by A1,GROUP_2:50;
  then (a" * b") * (a * b) in H by A2,GROUP_2:50;
  hence thesis by Th16;
end;
