reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th31:
  for a,b st a,b are_coprime & a + b > 2 holds
    a+b divides a|^n+b|^n iff not a+b divides a|^n-b|^n
  proof
    let a,b such that
    A1: a,b are_coprime & a + b > 2;
    A2: b > 0
    proof
      assume not thesis; then
      b = 0;
      hence contradiction by A1;
    end;
    A3: not (a+b) divides 2*a|^n by A1,A2,Lm1;
    (a+b) divides a|^n-b|^n implies not (a+b divides a|^n+b|^n)
    proof
      assume not thesis; then
      a+b divides (a|^n+b|^n +(a|^n -b|^n)) by WSIERP_1:4;
      hence contradiction by A3;
    end;
    hence thesis by NEWTON01:38;
  end;
