1637 = (2 * 818) + 1 ;
hence not 2 divides 1637 by NAT_4:9; :: thesis:
1637 = (3 * 545) + 2 ;
hence not 3 divides 1637 by NAT_4:9; :: thesis:
1637 = (5 * 327) + 2 ;
hence not 5 divides 1637 by NAT_4:9; :: thesis:
1637 = (7 * 233) + 6 ;
hence not 7 divides 1637 by NAT_4:9; :: thesis:
1637 = (11 * 148) + 9 ;
hence not 11 divides 1637 by NAT_4:9; :: thesis:
1637 = (13 * 125) + 12 ;
hence not 13 divides 1637 by NAT_4:9; :: thesis:
1637 = (17 * 96) + 5 ;
hence not 17 divides 1637 by NAT_4:9; :: thesis:
1637 = (19 * 86) + 3 ;
hence not 19 divides 1637 by NAT_4:9; :: thesis:
1637 = (23 * 71) + 4 ;
hence not 23 divides 1637 by NAT_4:9; :: thesis:
1637 = (29 * 56) + 13 ;
hence not 29 divides 1637 by NAT_4:9; :: thesis:
1637 = (31 * 52) + 25 ;
hence not 31 divides 1637 by NAT_4:9; :: thesis:
1637 = (37 * 44) + 9 ;
hence not 37 divides 1637 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1637 & n is prime holds
not n divides 1637 by XPRIMET1:24;
hence 1637 is prime by NAT_4:14; :: thesis: