reserve A for Tolerance_Space,
  X, Y for Subset of A;
reserve A for Approximation_Space,
  X for Subset of A;
reserve A for finite Tolerance_Space,
  X for Subset of A,
  x for Element of A;
reserve A for finite Approximation_Space,
  X, Y for Subset of A,
  x for Element of A;

theorem
  BndAp X = { x where x is Element of A : 0 < MemberFunc (X, A).x &
  MemberFunc (X, A).x < 1 }
proof
  thus BndAp X c= { x where x is Element of A : 0 < MemberFunc (X, A).x &
  MemberFunc (X, A).x < 1 }
  proof
    let y be object;
    assume
A1: y in BndAp X;
    then reconsider y9 = y as Element of A;
    0 < MemberFunc (X, A).y9 & MemberFunc (X, A).y9 < 1 by A1,Th42;
    hence thesis;
  end;
  let y be object;
  assume y in { x where x is Element of A : 0 < MemberFunc (X, A).x &
  MemberFunc (X, A).x < 1 };
  then ex y9 being Element of A st y9 = y & 0 < MemberFunc (X, A).y9 &
  MemberFunc (X, A).y9 < 1;
  hence thesis by Th42;
end;
