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;
reserve f for Ordinal-Sequence;

theorem Th22:
  f = numbering X & x in dom f & y in dom f implies (x c= y iff f.x c= f.y)
  proof assume
A1: f = numbering X & x in dom f & y in dom f; then
    dom f = ord-type X &
    f is_isomorphism_of RelIncl ord-type X, RelIncl On X by Th18,Th21;
    hence thesis by A1,Th6;
  end;
