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) = N `(N ~ A)
proof
  thus N ~ (N ~ A) c= N `(N ~ A)
  proof
    let x be object;
    assume
A1: x in N ~ (N ~ A);
    then reconsider x as Element of G;
    x * N meets N ~ A by A1,Th32;
    then consider z being object such that
A2: z in x * N & z in N ~ A by XBOOLE_0:3;
    reconsider z as Element of G by A2;
    z * N meets A by A2,Th14;
    then
A3: x * N meets A by A2,Th2;
  x * N c= N ~ A
  proof
    let y be object;
    assume
A4: y in x * N;
    then reconsider y as Element of G;
    x * N = y * N by A4,Th2;
    hence thesis by A3;
  end;
  hence thesis;
end;
    let x be object;
    assume
A5: x in N ` (N ~ A);
    then reconsider x as Element of G;
A6: x * N c= N ~ A by A5,Th31;
    x in x * N by GROUP_2:108;
    then x * N meets N ~ A by A6,XBOOLE_0:3;
    hence thesis;
end;
