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 ~ (N ` A)) = N ` A
proof
  thus  N ` (N ~ (N ` A)) c= N ` A
  proof
    let x be object;
    assume x in N ` (N ~ (N ` A));
    then consider x1 being Element of G such that
A1: x1 = x & x1 * N c= N ~ (N ` A);
    x1 in x1 * N by GROUP_2:108;
    then x1 * N meets N ` A by A1,Th33;
    then consider y be object such that
A2: y in x1 * N & y in N ` A by XBOOLE_0:3;
    reconsider y as Element of G by A2;
    y * N c= A by A2,Th12;
    then x1 * N c= A by A2,Th2;
    hence thesis by A1;
  end;
  let x be object;
  assume
A3: x in N ` A;
  then reconsider x as Element of G;
  x * N c= N ~ (N ` A)
  proof
    let y be object;
    assume
A4: y in x * N;
    then reconsider y as Element of G;
    y * N = x * N by A4,Th2;
    then x in y * N by GROUP_2:108;
    then y * N meets N ` A by A3,XBOOLE_0:3;
    hence thesis;
  end;
  hence thesis;
end;
