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;
A2: x * N c= N ` A by A1,Th41;
    x in x * N by GROUP_2:108;
    then x * N meets N ` A by A2,XBOOLE_0:3;
    hence thesis;
  end;
  let x be object;
  assume
A3:x in N ~ (N ` A);
  then reconsider x as Element of G;
  x * N meets N ` A by A3,Th33;
  then consider z being object such that
A4:z in x * N & z in N ` A by XBOOLE_0:3;
   reconsider z as Element of G by A4;
   z * N c= A by A4,Th12;
  then
A5: x * N c= A by A4,Th2;
  x * N c= N ` A
  proof
    let y be object;
    assume
A6: y in x * N;
    then reconsider y as Element of G;
    x * N = y * N by A6,Th2;
    hence thesis by A5;
  end;
  hence thesis;
end;
