reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th57:
  n <> 0 implies 3,Fermat(n) are_coprime
proof
  assume
A1: n <> 0;
  Fermat(n) gcd 3 <> 3
  proof
    assume Fermat(n) gcd 3 = 3;
    then 3 divides Fermat(n) by NAT_D:def 5;
    then consider k being Element of NAT such that
A2: 3 = k*(2 |^ (n+1)) + 1 by Th41,Th56;
    1 = k*(2 |^ (n+1)) div 2 by A2;
    then 1 = k*(2 |^ n) by Lm5;
    then 2 |^ n = 1 by NAT_1:15;
    then 2 |^ n = 2 |^ 0 by NEWTON:4;
    hence contradiction by A1,Th30;
  end;
  hence thesis by Th2,Th41;
end;
