let A be Approximation_Space; :: thesis: for X being Subset of A holds UAp (UAp X) = LAp (UAp X)
let X be Subset of A; :: thesis: UAp (UAp X) = LAp (UAp X)
thus UAp (UAp X) c= LAp (UAp X) :: according to XBOOLE_0:def 10 :: thesis: LAp (UAp X) c= UAp (UAp X)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in UAp (UAp X) or x in LAp (UAp X) )
assume A1: x in UAp (UAp X) ; :: thesis: x in LAp (UAp X)
then Class the InternalRel of A,x meets UAp X by Th10;
then consider z being set such that
A2: ( z in Class the InternalRel of A,x & z in UAp X ) by XBOOLE_0:3;
A3: Class the InternalRel of A,z meets X by A2, Th10;
A4: Class the InternalRel of A,z = Class the InternalRel of A,x by A1, A2, EQREL_1:31;
Class the InternalRel of A,x c= UAp X

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Class the InternalRel of A,x or y in UAp X )
assume A5: y in Class the InternalRel of A,x ; :: thesis: y in UAp X
then Class the InternalRel of A,x = Class the InternalRel of A,y by A1, EQREL_1:31;
hence y in UAp X by A3, A4, A5; :: thesis: verum

end;

hence x in LAp (UAp X) by A1; :: thesis: verum

end;

thus LAp (UAp X) c= UAp (UAp X) by Th14; :: thesis: verum