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
  N ~ (N ` A) c= A
proof
  let x be object;
  assume
A1:x in N ~ (N ` A);
  then reconsider x9 = x as Element of G;
  x9 * N meets N ` A by A1,Th33;
  then consider y being object such that
A2:y in x9 * N & y in N ` A by XBOOLE_0:3;
  reconsider y9 = y as Element of G by A2;
  y9 * N c= A by A2,Th12; then
A3:x9 * N c= A by A2,Th2;
  x9 in x9 * N by GROUP_2:108;
  hence thesis by A3;
end;
