3769 = (2 * 1884) + 1 ;
hence not 2 divides 3769 by NAT_4:9; :: thesis:
3769 = (3 * 1256) + 1 ;
hence not 3 divides 3769 by NAT_4:9; :: thesis:
3769 = (5 * 753) + 4 ;
hence not 5 divides 3769 by NAT_4:9; :: thesis:
3769 = (7 * 538) + 3 ;
hence not 7 divides 3769 by NAT_4:9; :: thesis:
3769 = (11 * 342) + 7 ;
hence not 11 divides 3769 by NAT_4:9; :: thesis:
3769 = (13 * 289) + 12 ;
hence not 13 divides 3769 by NAT_4:9; :: thesis:
3769 = (17 * 221) + 12 ;
hence not 17 divides 3769 by NAT_4:9; :: thesis:
3769 = (19 * 198) + 7 ;
hence not 19 divides 3769 by NAT_4:9; :: thesis:
3769 = (23 * 163) + 20 ;
hence not 23 divides 3769 by NAT_4:9; :: thesis:
3769 = (29 * 129) + 28 ;
hence not 29 divides 3769 by NAT_4:9; :: thesis:
3769 = (31 * 121) + 18 ;
hence not 31 divides 3769 by NAT_4:9; :: thesis:
3769 = (37 * 101) + 32 ;
hence not 37 divides 3769 by NAT_4:9; :: thesis:
3769 = (41 * 91) + 38 ;
hence not 41 divides 3769 by NAT_4:9; :: thesis:
3769 = (43 * 87) + 28 ;
hence not 43 divides 3769 by NAT_4:9; :: thesis:
3769 = (47 * 80) + 9 ;
hence not 47 divides 3769 by NAT_4:9; :: thesis:
3769 = (53 * 71) + 6 ;
hence not 53 divides 3769 by NAT_4:9; :: thesis:
3769 = (59 * 63) + 52 ;
hence not 59 divides 3769 by NAT_4:9; :: thesis:
3769 = (61 * 61) + 48 ;
hence not 61 divides 3769 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3769 & n is prime holds
not n divides 3769 by XPRIMET1:36;
hence 3769 is prime by NAT_4:14; :: thesis: