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 Th40:
  MemberFunc (X, A).x = 1 iff x in LAp X
proof
  hereby
    assume
A1: MemberFunc (X, A).x = 1;
    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 CARD_2:102,XBOOLE_1:17;
    then Class (the InternalRel of A, x) c= X by XBOOLE_1:18;
    hence x in LAp X;
  end;
  assume x in LAp X;
  then
A2: 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 thesis by A2,XCMPLX_1:60;
end;
