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 a be non zero square Nat, p be prime Nat st
  p divides a holds not a+p is square
proof
  let a be non zero square Nat;
  let p be prime Nat;
  assume
  B0: p divides a; then
  consider k such that
  B1: a = p*k by NAT_D:def 3;
  B3:  not p is trivial;
  p^2 divides a by B0,LmSQ; then
  p*p divides p*k by B1,SQUARE_1:def 1; then
  p divides k by INT_4:7; then
  consider n such that
  B4: k = p*n by NAT_D:def 3;
  not p divides 1*(k+1) by B3,INT_2:13,B4,NEWTON02:77; then
  not p*p divides p*(k+1) by INT_4:7; then
  not p^2 divides p*(k+1) by SQUARE_1:def 1;
  hence thesis by B1,LmSQ,INT_1:def 3;
end;
