theorem
  {x} c= O implies canonical_isomorphism_of (RelIncl order_type_of
  RelIncl {x}, RelIncl {x}) = 0 .--> x
proof
  set X = {x};
  set R = RelIncl X;
  set C = canonical_isomorphism_of (RelIncl order_type_of R,R);
A1: RelIncl {0} = {[0,0]} by WELLORD2:22;
  assume
A2: X c= O;
  then
A3: order_type_of R = {0} by Th36,CARD_1:49;
  R is well-ordering by A2,WELLORD2:8;
  then R, RelIncl order_type_of R are_isomorphic by WELLORD2:def 2;
  then
A4: RelIncl order_type_of R, R are_isomorphic by WELLORD1:40;
  RelIncl order_type_of R is well-ordering by WELLORD2:8;
  then
A5: C is_isomorphism_of RelIncl order_type_of R, R by A4,WELLORD1:def 9;
  then
A6: rng canonical_isomorphism_of(RelIncl {0}, R) = field R by A3,WELLORD1:def 7
    .= X by WELLORD2:def 1;
  dom canonical_isomorphism_of(RelIncl {0}, R) = field RelIncl {0} by A5,A3,
WELLORD1:def 7
    .= {0} by A1,RELAT_1:173;
  hence thesis by A3,A6,FUNCT_4:112;
end;
