 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;
 reserve X for Tolerance_Space,
         A, B, C for RoughSet of X;
 reserve K,L,M for Element of RoughSets X;

theorem Th71:
  for X being Tolerance_Space, A, B being Element of RSLattice X,
      A9, B9 being RoughSet of X st A = A9 & B = B9 holds
    A [= B iff
      LAp A9 c= LAp B9 & UAp A9 c= UAp B9
  proof
    let X be Tolerance_Space,
        A, B be Element of RSLattice X,
        A9, B9 be RoughSet of X;
    assume
A1: A = A9 & B = B9;
A2: A is Element of RoughSets X & B is Element of RoughSets X by Def23;
    thus A [= B implies LAp A9 c= LAp B9 & UAp A9 c= UAp B9
    proof
      assume A [= B; then
      A "\/" B = B; then
      A9 _\/_ B9 = B9 by A1,Def23,A2; then
      LAp A9 \/ LAp B9 = LAp B9 & UAp A9 \/ UAp B9 = UAp B9;
      hence thesis by XBOOLE_1:11;
    end;
    assume LAp A9 c= LAp B9 & UAp A9 c= UAp B9; then
    LAp A9 \/ LAp B9 = LAp B9 & UAp A9 \/ UAp B9 = UAp B9 by XBOOLE_1:12; then
    LAp (A9 _\/_ B9) = LAp B9 & UAp (A9 _\/_ B9) = UAp B9; then
A3: A9 _\/_ B9 = B9;
    reconsider A1 = A, B1 = B as Element of RoughSets X by Def23;
    reconsider A9 = A1, B9 = B1 as RoughSet of X by Def20;
    A9 _\/_ B9 = A "\/" B by Def23;
    hence thesis by A3,A1;
  end;
