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

theorem :: Problem 66
  4,11 are_coprime &
  for m being Nat holds not ArProg (4,11).m is Fibonacci
proof
A1:  4 gcd 11 = 4 gcd (11-4) = 4 gcd (7-4) by PREPOWER:97;
  3 gcd 4 = 3 gcd (4-3) by PREPOWER:97;
  hence 4,11 are_coprime by A1;
A2: 0*11+0 mod 11 = 0 & 5*11+0 mod 11 = 0 & 11*0+1 mod 11 = 1
    & 11*8+1 mod 11 = 1 by NUMBER02:16;
  then Fib(0),Fib(0+10) are_congruent_mod 11 &
    Fib(0+1),Fib(0+10+1) are_congruent_mod 11 by Th7,PRE_FF:1,NAT_D:64;
  then
A3: (for x,y be Nat st x,y are_congruent_mod 10 holds
       Fib(x),Fib(y) are_congruent_mod 11) by Th11;
  for x be Nat st x < 10 holds Fib(x) mod 11 <> 4
  proof
    let x be Nat such that
A4:   x < 10;
A5:   x=0 or ... or x=10 by A4;
    0*11+2 mod 11 = 2 & 11*0+3 mod 11 = 3 & 0*11+5 mod 11 = 5 &
    11*0+8 mod 11 = 8 & 1*11+2 mod 11 = 2 & 11*1+10 mod 11 = 10 &
    3*11+1 mod 11 = 1 by NUMBER02:16;
    hence thesis by A5,A2,Th7,FIB_NUM2:21,22,23,PRE_FF:1,NUMBER06:46;
  end;
  hence thesis by A3,Th12;
end;
