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 Th41:
  MemberFunc (X, A).x = 0 iff x in (UAp X)`
proof
  hereby
    assume
A1: MemberFunc (X, A).x = 0;
    MemberFunc (X, A).x = card (X /\ Class (the InternalRel of A, x)) / (
    card Class (the InternalRel of A, x)) by Def9;
    then X /\ Class (the InternalRel of A, x) = {} by A1,XCMPLX_1:50;
    then X misses Class (the InternalRel of A, x);
    then not x in UAp X by Th10;
    hence x in (UAp X)` by XBOOLE_0:def 5;
  end;
  assume x in (UAp X)`;
  then not x in UAp X by XBOOLE_0:def 5;
  then X misses Class (the InternalRel of A, x);
  then
A2: card (X /\ Class (the InternalRel of A, x)) = 0;
  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;
end;
