
theorem Th9:
  for R being non empty RelStr, x being Element of R st (the
  InternalRel of R)-Seg x c= well_founded-Part R holds x in well_founded-Part R
proof
  let R be non empty RelStr, x be Element of R;
  set wfp = well_founded-Part R, IT = {S where S is Subset of R : S is
  well_founded lower}, xwfp = wfp \/ {x};
A1: wfp = union IT by Def4;
  x in {x} by TARSKI:def 1;
  then
A2: x in xwfp by XBOOLE_0:def 3;
  reconsider xwfp as Subset of R;
  {x} is well_founded Subset of R by Th6;
  then
A3: xwfp is well_founded by Th7;
  assume (the InternalRel of R)-Seg x c= wfp;
  then xwfp is lower by Th5;
  then xwfp in IT by A3;
  hence thesis by A1,A2,TARSKI:def 4;
end;
