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 Th39:
  A c= N ` (N ~ A) implies N ~ A c= N ~ (N `(N ~ A))
proof
  assume
A1: A c= N ` (N ~ A);
  let x be object;
  assume
A2: x in N ~ A;
  then reconsider x as Element of G;
  x * N meets A by A2,Th14;
  then x * N meets N ` (N ~ A) by A1,XBOOLE_1:63;
  hence thesis;
end;
