let A be Approximation_Space; :: thesis: for X being Subset of A holds LAp (LAp X) = LAp X
let X be Subset of A; :: thesis: LAp (LAp X) = LAp X
thus LAp (LAp X) c= LAp X by Th12; :: according to XBOOLE_0:def 10 :: thesis: LAp X c= LAp (LAp X)
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LAp X or x in LAp (LAp X) )
assume A1: x in LAp X ; :: thesis: x in LAp (LAp X)
then A2: Class the InternalRel of A,x c= X by Th8;
Class the InternalRel of A,x c= LAp X

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Class the InternalRel of A,x or y in LAp X )
assume A3: y in Class the InternalRel of A,x ; :: thesis: y in LAp X
then Class the InternalRel of A,x = Class the InternalRel of A,y by A1, EQREL_1:31;
hence y in LAp X by A2, A3; :: thesis: verum

end;

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