3637 = (2 * 1818) + 1 ;
hence not 2 divides 3637 by NAT_4:9; :: thesis:
3637 = (3 * 1212) + 1 ;
hence not 3 divides 3637 by NAT_4:9; :: thesis:
3637 = (5 * 727) + 2 ;
hence not 5 divides 3637 by NAT_4:9; :: thesis:
3637 = (7 * 519) + 4 ;
hence not 7 divides 3637 by NAT_4:9; :: thesis:
3637 = (11 * 330) + 7 ;
hence not 11 divides 3637 by NAT_4:9; :: thesis:
3637 = (13 * 279) + 10 ;
hence not 13 divides 3637 by NAT_4:9; :: thesis:
3637 = (17 * 213) + 16 ;
hence not 17 divides 3637 by NAT_4:9; :: thesis:
3637 = (19 * 191) + 8 ;
hence not 19 divides 3637 by NAT_4:9; :: thesis:
3637 = (23 * 158) + 3 ;
hence not 23 divides 3637 by NAT_4:9; :: thesis:
3637 = (29 * 125) + 12 ;
hence not 29 divides 3637 by NAT_4:9; :: thesis:
3637 = (31 * 117) + 10 ;
hence not 31 divides 3637 by NAT_4:9; :: thesis:
3637 = (37 * 98) + 11 ;
hence not 37 divides 3637 by NAT_4:9; :: thesis:
3637 = (41 * 88) + 29 ;
hence not 41 divides 3637 by NAT_4:9; :: thesis:
3637 = (43 * 84) + 25 ;
hence not 43 divides 3637 by NAT_4:9; :: thesis:
3637 = (47 * 77) + 18 ;
hence not 47 divides 3637 by NAT_4:9; :: thesis:
3637 = (53 * 68) + 33 ;
hence not 53 divides 3637 by NAT_4:9; :: thesis:
3637 = (59 * 61) + 38 ;
hence not 59 divides 3637 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3637 & n is prime holds
not n divides 3637 by XPRIMET1:34;
hence 3637 is prime by NAT_4:14; :: thesis: