
theorem Th21:
  for R being non empty RelStr, N being Subset of R, x being set st
  x in min-classes N holds x is non empty
proof
  let R be non empty RelStr, N be Subset of R, x be set;
  assume x in min-classes N;
  then consider y being Element of R\~ such that
A1: y is_minimal_wrt N, the InternalRel of R\~ and
A2: x = Class(EqRel R, y) /\ N by Def8;
A3: y in N by A1,WAYBEL_4:def 25;
  y in Class(EqRel R, y) by EQREL_1:20;
  hence thesis by A2,A3,XBOOLE_0:def 4;
end;
