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 Th35:
  for X being finite set st X c= O holds order_type_of RelIncl X =
  card X
proof
  let X be finite set;
  assume
A1: X c= O;
  then order_type_of RelIncl X is finite by Lm4;
  then reconsider o = order_type_of RelIncl X as Nat;
  card X = card order_type_of RelIncl X by A1,Th7;
  then o = card X;
  hence thesis;
end;
