reserve a,b,d,n,k,i,j,x,s for Nat;

theorem Th12:
  for i,j,k being Nat st 0 < j & k < i &
     (for x,y being Nat st x,y are_congruent_mod j holds
       Fib(x),Fib(y) are_congruent_mod i) &
      (for x be Nat st x < j holds Fib(x) mod i <> k)
  holds
    for m be Nat holds not ArProg (k,i).m is Fibonacci
proof
  let i,j,k be Nat such that
A1:  0 < j & k < i and
A2:  for x,y be Nat st x,y are_congruent_mod j holds
       Fib(x),Fib(y) are_congruent_mod i and
A3:  for x be Nat st x < j holds Fib(x) mod i <> k;
  let m be Nat;
  assume ArProg (k,i).m is Fibonacci;
  then consider f be Nat such that
A4: ArProg (k,i).m= Fib(f);
  ArProg (k,i).m = k + i*m by NUMBER06:7;
  then
A5: Fib(f) mod i = k by A1,A4,NUMBER02:16;
A6: Fib(f mod j) mod i <>k by A3, A1,INT_1:58;
  Fib(f),Fib(f mod j) are_congruent_mod i by A2,A1,NUMPOLY1:4;
  hence thesis by A5,A6,NAT_D:64;
end;
