theorem Th5:
  for N,T
  for k1, k2 being Nat st il.(T,k1) = il.(T,k2) holds
  k1 = k2
proof let N,T;
  let k1, k2 be Nat;
  assume
A1: il.(T,k1) = il.(T,k2);
A2: k1 is Element of NAT & k2 is Element of NAT by ORDINAL1:def 12;
  consider f2 being sequence of NAT such that
A3: f2 is bijective & for m, n being Element of NAT holds m <= n iff f2.
  m <= f2. n, T and
A4: il.(T,k2) = f2.k2 by Def4;
  consider f1 being sequence of NAT such that
A5: f1 is bijective and
A6: for m, n being Element of NAT holds m <= n iff f1.m <= f1.n, T and
A7: il.(T,k1) = f1.k1 by Def4;
A8: dom f1 = NAT by FUNCT_2:def 1;
  f1 = f2 by A5,A6,A3,Th2;
  hence thesis by A1,A2,A5,A7,A4,A8,FUNCT_1:def 4;
end;
