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;

theorem Th7:
  X c= A implies card X = card order_type_of RelIncl X
proof
  set R = RelIncl X;
  set B = order_type_of R;
  set phi = canonical_isomorphism_of (RelIncl B,R);
  assume X c= A;
  then R is well-ordering by WELLORD2:8;
  then R, RelIncl B are_isomorphic by WELLORD2:def 2;
  then RelIncl B is well-ordering & RelIncl B, R are_isomorphic by WELLORD1:40
,WELLORD2:8;
  then
A1: phi is_isomorphism_of RelIncl B,R by WELLORD1:def 9;
  field R = X by WELLORD2:def 1;
  then
A2: rng phi = X by A1,WELLORD1:def 7;
  field RelIncl B = B by WELLORD2:def 1;
  then
A3: dom phi = B by A1,WELLORD1:def 7;
  phi is one-to-one by A1,WELLORD1:def 7;
  then B,X are_equipotent by A3,A2,WELLORD2:def 4;
  hence thesis by CARD_1:5;
end;
