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
  for A being finite Approximation_Space, X being Subset of A,
      x, y being set st [x,y] in the InternalRel of A holds
     MemberFunc (X, A).x = MemberFunc (X, A).y
proof
  let A be finite Approximation_Space, X be Subset of A, x, y be set;
  assume
A1: [x,y] in the InternalRel of A;
  then
A2: y is Element of A by ZFMISC_1:87;
  x is Element of A by A1,ZFMISC_1:87;
  then
A3: MemberFunc (X, A).x = card (X /\ Class (the InternalRel of A, x)) / (
  card Class (the InternalRel of A, x)) by Def9;
  x in Class (the InternalRel of A, y) by A1,EQREL_1:19;
  then
  Class (the InternalRel of A, x) = Class (the InternalRel of A, y) by A2,
EQREL_1:23;
  hence thesis by A2,A3,Def9;
end;
