let A be Approximation_Space; :: thesis: for X being Subset of A holds UAp (UAp X) = UAp X
let X be Subset of A; :: thesis: UAp (UAp X) = UAp X

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: UAp X c= UAp (UAp X)

let x be set ; :: thesis: ( x in UAp (UAp X) implies x in UAp X )
assume A1: x in UAp (UAp X) ; :: thesis: x in UAp X
then Class the InternalRel of A,x meets UAp X by Th10;
then consider y being set such that
A2: ( y in Class the InternalRel of A,x & y in UAp X ) by XBOOLE_0:3;
A3: Class the InternalRel of A,y meets X by A2, Th10;
Class the InternalRel of A,y = Class the InternalRel of A,x by A1, A2, EQREL_1:31;
hence x in UAp X by A1, A3; :: thesis: verum

end;

thus UAp X c= UAp (UAp X) by Th13; :: thesis: verum