reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  gr(A /\ B) is Subgroup of gr A /\ gr B
proof
  now
    let a;
    assume a in gr(A /\ B);
    then consider F,I such that
A1: len F = len I and
A2: rng F c= A /\ B and
A3: Product(F |^ I) = a by Th28;
    A /\ B c= B by XBOOLE_1:17;
    then rng F c= B by A2;
    then
A4: a in gr B by A1,A3,Th28;
    A /\ B c= A by XBOOLE_1:17;
    then rng F c= A by A2;
    then a in gr A by A1,A3,Th28;
    hence a in gr A /\ gr B by A4,GROUP_2:82;
  end;
  hence thesis by GROUP_2:58;
end;
