3793 = (2 * 1896) + 1 ;
hence not 2 divides 3793 by NAT_4:9; :: thesis:
3793 = (3 * 1264) + 1 ;
hence not 3 divides 3793 by NAT_4:9; :: thesis:
3793 = (5 * 758) + 3 ;
hence not 5 divides 3793 by NAT_4:9; :: thesis:
3793 = (7 * 541) + 6 ;
hence not 7 divides 3793 by NAT_4:9; :: thesis:
3793 = (11 * 344) + 9 ;
hence not 11 divides 3793 by NAT_4:9; :: thesis:
3793 = (13 * 291) + 10 ;
hence not 13 divides 3793 by NAT_4:9; :: thesis:
3793 = (17 * 223) + 2 ;
hence not 17 divides 3793 by NAT_4:9; :: thesis:
3793 = (19 * 199) + 12 ;
hence not 19 divides 3793 by NAT_4:9; :: thesis:
3793 = (23 * 164) + 21 ;
hence not 23 divides 3793 by NAT_4:9; :: thesis:
3793 = (29 * 130) + 23 ;
hence not 29 divides 3793 by NAT_4:9; :: thesis:
3793 = (31 * 122) + 11 ;
hence not 31 divides 3793 by NAT_4:9; :: thesis:
3793 = (37 * 102) + 19 ;
hence not 37 divides 3793 by NAT_4:9; :: thesis:
3793 = (41 * 92) + 21 ;
hence not 41 divides 3793 by NAT_4:9; :: thesis:
3793 = (43 * 88) + 9 ;
hence not 43 divides 3793 by NAT_4:9; :: thesis:
3793 = (47 * 80) + 33 ;
hence not 47 divides 3793 by NAT_4:9; :: thesis:
3793 = (53 * 71) + 30 ;
hence not 53 divides 3793 by NAT_4:9; :: thesis:
3793 = (59 * 64) + 17 ;
hence not 59 divides 3793 by NAT_4:9; :: thesis:
3793 = (61 * 62) + 11 ;
hence not 61 divides 3793 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3793 & n is prime holds
not n divides 3793 by XPRIMET1:36;
hence 3793 is prime by NAT_4:14; :: thesis: