let A be finite Approximation_Space; :: thesis: for X being Subset of A
for x being Element of A holds
( (MemberFunc X,A) . x = 1 iff x in LAp X )
let X be Subset of A; :: thesis: for x being Element of A holds
( (MemberFunc X,A) . x = 1 iff x in LAp X )
let x be Element of A; :: thesis: ( (MemberFunc X,A) . x = 1 iff x in LAp X )

hereby :: thesis: ( x in LAp X implies (MemberFunc X,A) . x = 1 )

assume A1: (MemberFunc X,A) . x = 1 ; :: thesis: x in LAp X
(MemberFunc X,A) . x = (card (X /\ (Class the InternalRel of A,x))) / (card (Class the InternalRel of A,x)) by Def9;
then card (X /\ (Class the InternalRel of A,x)) = card (Class the InternalRel of A,x) by A1, XCMPLX_1:58;
then X /\ (Class the InternalRel of A,x) = Class the InternalRel of A,x by TRIANG_1:3, XBOOLE_1:17;
then Class the InternalRel of A,x c= X by XBOOLE_1:18;
hence x in LAp X ; :: thesis: verum

end;

assume x in LAp X ; :: thesis: (MemberFunc X,A) . x = 1
then A3: card (X /\ (Class the InternalRel of A,x)) = card (Class the InternalRel of A,x) by Th8, XBOOLE_1:28;
(MemberFunc X,A) . x = (card (X /\ (Class the InternalRel of A,x))) / (card (Class the InternalRel of A,x)) by Def9;
hence (MemberFunc X,A) . x = 1 by A3, XCMPLX_1:60; :: thesis: verum