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;
