reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  for p be prime Nat, k be non zero Nat st k < p holds
    not p divides p|^2 - k|^2
  proof
    let p be prime Nat, k be non zero Nat;
    reconsider a = k-1 as Nat;
    p divides (p - (a+1))*(p + (a+1)) implies a+1 >= p by PSQ;
    hence thesis by NEWTON01:1;
  end;
