reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;
reserve a,b for Aleph;
reserve O for Ordinal,
        F for Subset of omega;

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;
