reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;
reserve e,u for set;

theorem
  for a,b being Ordinal st a*^b is natural non empty holds a in omega &
  b in omega
proof
  let x,y be Ordinal such that
A1: x*^y in omega;
  assume
A2: x*^y is non empty;
  then y <> {} by ORDINAL2:38;
  then
A3: x c= x*^y by Th36;
  x <> {} by A2,ORDINAL2:35;
  then y c= x*^y by Th36;
  hence thesis by A1,A3,ORDINAL1:12;
end;
