4637 = (2 * 2318) + 1 ;
hence not 2 divides 4637 by NAT_4:9; :: thesis:
4637 = (3 * 1545) + 2 ;
hence not 3 divides 4637 by NAT_4:9; :: thesis:
4637 = (5 * 927) + 2 ;
hence not 5 divides 4637 by NAT_4:9; :: thesis:
4637 = (7 * 662) + 3 ;
hence not 7 divides 4637 by NAT_4:9; :: thesis:
4637 = (11 * 421) + 6 ;
hence not 11 divides 4637 by NAT_4:9; :: thesis:
4637 = (13 * 356) + 9 ;
hence not 13 divides 4637 by NAT_4:9; :: thesis:
4637 = (17 * 272) + 13 ;
hence not 17 divides 4637 by NAT_4:9; :: thesis:
4637 = (19 * 244) + 1 ;
hence not 19 divides 4637 by NAT_4:9; :: thesis:
4637 = (23 * 201) + 14 ;
hence not 23 divides 4637 by NAT_4:9; :: thesis:
4637 = (29 * 159) + 26 ;
hence not 29 divides 4637 by NAT_4:9; :: thesis:
4637 = (31 * 149) + 18 ;
hence not 31 divides 4637 by NAT_4:9; :: thesis:
4637 = (37 * 125) + 12 ;
hence not 37 divides 4637 by NAT_4:9; :: thesis:
4637 = (41 * 113) + 4 ;
hence not 41 divides 4637 by NAT_4:9; :: thesis:
4637 = (43 * 107) + 36 ;
hence not 43 divides 4637 by NAT_4:9; :: thesis:
4637 = (47 * 98) + 31 ;
hence not 47 divides 4637 by NAT_4:9; :: thesis:
4637 = (53 * 87) + 26 ;
hence not 53 divides 4637 by NAT_4:9; :: thesis:
4637 = (59 * 78) + 35 ;
hence not 59 divides 4637 by NAT_4:9; :: thesis:
4637 = (61 * 76) + 1 ;
hence not 61 divides 4637 by NAT_4:9; :: thesis:
4637 = (67 * 69) + 14 ;
hence not 67 divides 4637 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4637 & n is prime holds
not n divides 4637 by XPRIMET1:38;
hence 4637 is prime by NAT_4:14; :: thesis: