4363 = (2 * 2181) + 1 ;
hence not 2 divides 4363 by NAT_4:9; :: thesis:
4363 = (3 * 1454) + 1 ;
hence not 3 divides 4363 by NAT_4:9; :: thesis:
4363 = (5 * 872) + 3 ;
hence not 5 divides 4363 by NAT_4:9; :: thesis:
4363 = (7 * 623) + 2 ;
hence not 7 divides 4363 by NAT_4:9; :: thesis:
4363 = (11 * 396) + 7 ;
hence not 11 divides 4363 by NAT_4:9; :: thesis:
4363 = (13 * 335) + 8 ;
hence not 13 divides 4363 by NAT_4:9; :: thesis:
4363 = (17 * 256) + 11 ;
hence not 17 divides 4363 by NAT_4:9; :: thesis:
4363 = (19 * 229) + 12 ;
hence not 19 divides 4363 by NAT_4:9; :: thesis:
4363 = (23 * 189) + 16 ;
hence not 23 divides 4363 by NAT_4:9; :: thesis:
4363 = (29 * 150) + 13 ;
hence not 29 divides 4363 by NAT_4:9; :: thesis:
4363 = (31 * 140) + 23 ;
hence not 31 divides 4363 by NAT_4:9; :: thesis:
4363 = (37 * 117) + 34 ;
hence not 37 divides 4363 by NAT_4:9; :: thesis:
4363 = (41 * 106) + 17 ;
hence not 41 divides 4363 by NAT_4:9; :: thesis:
4363 = (43 * 101) + 20 ;
hence not 43 divides 4363 by NAT_4:9; :: thesis:
4363 = (47 * 92) + 39 ;
hence not 47 divides 4363 by NAT_4:9; :: thesis:
4363 = (53 * 82) + 17 ;
hence not 53 divides 4363 by NAT_4:9; :: thesis:
4363 = (59 * 73) + 56 ;
hence not 59 divides 4363 by NAT_4:9; :: thesis:
4363 = (61 * 71) + 32 ;
hence not 61 divides 4363 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4363 & n is prime holds
not n divides 4363 by XPRIMET1:36;
hence 4363 is prime by NAT_4:14; :: thesis: