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 Th35:
  N ~ A = N ~ (N ~ A)
proof
  thus N ~ A c= N ~ (N ~ A)
  proof
    let x be object;
    assume
A1: x in N ~ A;
    then reconsider x as Element of G;
    x in x * N by GROUP_2:108;
    then x * N meets N ~ A by A1,XBOOLE_0:3;
    hence thesis;
  end;
  let x be object;
  assume
A2:x in N ~ (N ~ A);
  then reconsider x9 = x as Element of G;
  x9 * N meets N ~ A by A2,Th32;
  then consider y be object such that
A3:y in x9 * N & y in N ~ A by XBOOLE_0:3;
   reconsider y9 = y as Element of G by A3;
A4:y9 * N meets A by A3,Th14;
  y9 * N = x9 * N by A3,Th2;
  hence thesis by A4;
end;
