reserve A for Tolerance_Space,
  X, Y for Subset of A;
reserve A for Approximation_Space,
  X for Subset of A;

theorem Th37:
  for A being Approximation_Space, X being Subset of A, x, y being
  set st x in UAp X & [x,y] in the InternalRel of A holds y in UAp X
proof
  let A be Approximation_Space, X be Subset of A;
  let x, y be set;
  assume that
A1: x in UAp X and
A2: [x,y] in the InternalRel of A;
  [y,x] in the InternalRel of A by A2,EQREL_1:6;
  then y in Class (the InternalRel of A, x) by EQREL_1:19;
  then
A3: Class (the InternalRel of A, x) = Class (the InternalRel of A, y) by A1,
EQREL_1:23;
  Class (the InternalRel of A, x) meets X & y is Element of A by A1,A2,Th10,
ZFMISC_1:87;
  hence thesis by A3;
end;
