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 Th23:
  f = numbering X & x in dom f & y in dom f implies (x in y iff f.x in f.y)
  proof assume
A1: f = numbering X & x in dom f & y in dom f; then
    y c= x iff f.y c= f.x by Th22;
    hence thesis by A1,Th4;
  end;
