
theorem Th6:
  for R being non empty RelStr, x be Element of R holds {x} is
  well_founded Subset of R
proof
  let R be non empty RelStr, x be Element of R;
  set r = the InternalRel of R;
  reconsider sx = {x} as Subset of R;
  sx is well_founded
  proof
    let Y be set;
    assume that
A1: Y c= sx and
A2: Y <> {};
    take x;
    Y = sx by A1,A2,ZFMISC_1:33;
    hence x in Y by TARSKI:def 1;
    assume not thesis;
    then consider a being object such that
A3: a in r-Seg x /\ Y by XBOOLE_0:4;
    a in r-Seg x by A3,XBOOLE_0:def 4;
    then
A4: a <> x by WELLORD1:1;
    a in Y by A3,XBOOLE_0:def 4;
    hence contradiction by A1,A4,TARSKI:def 1;
  end;
  hence thesis;
end;
