
theorem Th32: :: Corollary 4.44
  for R being non empty RelStr st R is quasi_ordered & R is Dickson
  holds R\~ is well_founded
proof
  let R be non empty RelStr such that
A1: R is quasi_ordered and
A2: R is Dickson;
  A3: for
 f being sequence of R ex i,j being Nat st i < j & f.i <= f.j
  by A2,Th28;
  now
    let N be Subset of R;
    assume N <> {};
    then min-classes N is non empty by A1,A3,Th30;
    hence ex x being object st x in min-classes N by XBOOLE_0:def 1;
  end;
  hence thesis by Th19;
end;
