
theorem
  for n being positive Integer holds n+1 divides n^2 + 1 iff n = 1
  proof
    let n be positive Integer;
    thus n+1 divides n^2 + 1 implies n = 1
    proof
      assume n+1 divides n^2 + 1;
      then n: n+1 divides n*(n+1)-(n-1);
      n+1 divides n*(n+1) by INT_1:def 3;
      then n1: n+1 divides n-1 by n,INT_5:2;
      assume o: n<>1;
      n >= 0+1 by INT_1:7;
      then n > 1 by o,XXREAL_0:1;
      then n-1 > 1-1 by XREAL_1:9;
      then n+1 <= n-1 by n1,INT_2:27;
      then n+1-n <= n-1-n by XREAL_1:9;
      hence contradiction;
    end;
    thus thesis;
  end;
