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 Th42:
  0 < MemberFunc (X, A).x & MemberFunc (X, A).x < 1 iff x in BndAp X
proof
  hereby
    assume that
A1: 0 < MemberFunc (X, A).x and
A2: MemberFunc (X, A).x < 1;
    not x in (UAp X)` by A1,Th41;
    then
A3: x in UAp X by XBOOLE_0:def 5;
    not x in LAp X by A2,Th40;
    hence x in BndAp X by A3,XBOOLE_0:def 5;
  end;
  assume
A4: x in BndAp X;
  then not x in LAp X by XBOOLE_0:def 5;
  then
A5: MemberFunc (X, A).x <> 1 by Th40;
  x in UAp X by A4,XBOOLE_0:def 5;
  then not x in (UAp X)` by XBOOLE_0:def 5;
  then
A6: 0 <> MemberFunc (X, A).x by Th41;
  MemberFunc (X, A).x <= 1 by Th38;
  hence thesis by A6,A5,Th38,XXREAL_0:1;
end;
