reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;

theorem Th18:
  dom numbering X = ord-type X & rng numbering X = On X
  proof
    set R1 = RelIncl ord-type X;
    set R2 = RelIncl On X;
    set f = canonical_isomorphism_of(R1, R2);
    consider a such that
A1: On X c= a by Def1;
    ex phi being Ordinal-Sequence st phi = f &
    phi is increasing & dom phi = ord-type X & rng phi = On X
    by A1,CARD_5:5;
    hence thesis;
  end;
