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

theorem Th13:
  Fib(0),Fib(12) are_congruent_mod 8 &
  Fib(1),Fib(12+1) are_congruent_mod 8 &
  for x be Nat st x < 12 holds Fib(x) mod 8 <> 4 & Fib(x) mod 8 <> 6
proof
  0*8+0 mod 8 = 0 & 18*8+0 mod 8 = 0 & 8*0+1 mod 8 = 1
    & 8*29+1 mod 8 = 1 by NUMBER02:16;
  hence Fib(0),Fib(12) are_congruent_mod 8 &
    Fib(1),Fib(12+1) are_congruent_mod 8 by Th7,PRE_FF:1,NAT_D:64;
  let x be Nat such that
A1:  x < 12;
A2:  x=0 or ... or x=12 by A1;
  0*8+0 mod 8 = 0 & 8*0+1 mod 8 = 1 & 0*8+2 mod 8 = 2 & 8*0+3 mod 8 = 3 &
  0*8+5 mod 8 = 5 & 8*1+0 mod 8 = 0 & 1*8+5 mod 8 = 5 & 8*2+5 mod 8 = 5 &
  3*8+5 mod 8 = 5 & 8*4+2 mod 8 = 2 & 8*6+7 mod 8 = 7 & 11*8+1 mod 8 = 1
    by NUMBER02:16;
  hence thesis by A1,A2,Th7,FIB_NUM2:21,22,23,PRE_FF:1,NUMBER06:46;
end;
