theorem Th6:
  X c= A implies sup X is_cofinal_with order_type_of RelIncl X
proof
  assume
A1: X c= A;
  then consider phi such that
  phi = canonical_isomorphism_of (RelIncl order_type_of RelIncl X, RelIncl
  X) and
A2: phi is increasing & dom phi = order_type_of RelIncl X & rng phi = X
  by Th5;
  take phi;
  On X = X by A1,ORDINAL3:6;
  hence thesis by A2,ORDINAL2:def 3;
end;
