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 Th34:
  N ` (N ` A) = N ` A
proof
  thus N ` (N ` A) c= N ` A
  proof
    let x be object;
    assume x in N ` (N ` A);
    then consider y being Element of G such that
A1: y = x & y * N c= N ` A;
    y in y * N  by GROUP_2:108;
    hence thesis by A1;
  end;
  let x be object;
  assume
A2: x in N ` A;
  then reconsider x9 = x as Element of G;
A3: x9 * N c= A by A2,Th12;
  x9 * N c= N ` A
  proof
    let y be object;
    assume
A4: y in x9 * N;
    then reconsider y9 = y as Element of G;
    x9 * N = y9 * N by A4,Th2;
    hence thesis by A3;
  end;
  hence thesis;
end;
