reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem
  for G be Group, A,B be non empty Subset of G, N be normal Subgroup of G
   holds N ` A * N ` B c= N ` (A * B)
proof
  let G be Group, A,B be non empty Subset of G,
  N be normal Subgroup of G;
  let x be object;
  assume
A1: x in N ` A * N ` B;
  then reconsider x as Element of G;
  consider x1, x2 be Element of G such that
A2: x = x1 * x2 & x1 in N ` A & x2 in N ` B by A1;
 x1 * N c= A & x2 * N c= B by A2,Th12;
  then (x1 * N) * (x2 * N) c= A * B by GROUP_3:4;
  then (x1 * x2) * N c= A * B by Th1;
  hence thesis by A2;
end;
