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
  LAp X = { x where x is Element of A : MemberFunc (X, A).x = 1 }
proof
  thus LAp X c= { x where x is Element of A : MemberFunc (X, A).x = 1 }
  proof
    let y be object;
    assume
A1: y in LAp X;
    then reconsider y9 = y as Element of A;
    MemberFunc (X, A).y9 = 1 by A1,Th40;
    hence thesis;
  end;
  let y be object;
  assume y in { x where x is Element of A : MemberFunc (X, A).x = 1 };
  then ex y9 being Element of A st y9 = y & MemberFunc (X, A). y9 = 1;
  hence thesis by Th40;
end;
