3673 = (2 * 1836) + 1 ;
hence not 2 divides 3673 by NAT_4:9; :: thesis:
3673 = (3 * 1224) + 1 ;
hence not 3 divides 3673 by NAT_4:9; :: thesis:
3673 = (5 * 734) + 3 ;
hence not 5 divides 3673 by NAT_4:9; :: thesis:
3673 = (7 * 524) + 5 ;
hence not 7 divides 3673 by NAT_4:9; :: thesis:
3673 = (11 * 333) + 10 ;
hence not 11 divides 3673 by NAT_4:9; :: thesis:
3673 = (13 * 282) + 7 ;
hence not 13 divides 3673 by NAT_4:9; :: thesis:
3673 = (17 * 216) + 1 ;
hence not 17 divides 3673 by NAT_4:9; :: thesis:
3673 = (19 * 193) + 6 ;
hence not 19 divides 3673 by NAT_4:9; :: thesis:
3673 = (23 * 159) + 16 ;
hence not 23 divides 3673 by NAT_4:9; :: thesis:
3673 = (29 * 126) + 19 ;
hence not 29 divides 3673 by NAT_4:9; :: thesis:
3673 = (31 * 118) + 15 ;
hence not 31 divides 3673 by NAT_4:9; :: thesis:
3673 = (37 * 99) + 10 ;
hence not 37 divides 3673 by NAT_4:9; :: thesis:
3673 = (41 * 89) + 24 ;
hence not 41 divides 3673 by NAT_4:9; :: thesis:
3673 = (43 * 85) + 18 ;
hence not 43 divides 3673 by NAT_4:9; :: thesis:
3673 = (47 * 78) + 7 ;
hence not 47 divides 3673 by NAT_4:9; :: thesis:
3673 = (53 * 69) + 16 ;
hence not 53 divides 3673 by NAT_4:9; :: thesis:
3673 = (59 * 62) + 15 ;
hence not 59 divides 3673 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3673 & n is prime holds
not n divides 3673 by XPRIMET1:34;
hence 3673 is prime by NAT_4:14; :: thesis: