5923 = (2 * 2961) + 1 ;
hence not 2 divides 5923 by NAT_4:9; :: thesis:
5923 = (3 * 1974) + 1 ;
hence not 3 divides 5923 by NAT_4:9; :: thesis:
5923 = (5 * 1184) + 3 ;
hence not 5 divides 5923 by NAT_4:9; :: thesis:
5923 = (7 * 846) + 1 ;
hence not 7 divides 5923 by NAT_4:9; :: thesis:
5923 = (11 * 538) + 5 ;
hence not 11 divides 5923 by NAT_4:9; :: thesis:
5923 = (13 * 455) + 8 ;
hence not 13 divides 5923 by NAT_4:9; :: thesis:
5923 = (17 * 348) + 7 ;
hence not 17 divides 5923 by NAT_4:9; :: thesis:
5923 = (19 * 311) + 14 ;
hence not 19 divides 5923 by NAT_4:9; :: thesis:
5923 = (23 * 257) + 12 ;
hence not 23 divides 5923 by NAT_4:9; :: thesis:
5923 = (29 * 204) + 7 ;
hence not 29 divides 5923 by NAT_4:9; :: thesis:
5923 = (31 * 191) + 2 ;
hence not 31 divides 5923 by NAT_4:9; :: thesis:
5923 = (37 * 160) + 3 ;
hence not 37 divides 5923 by NAT_4:9; :: thesis:
5923 = (41 * 144) + 19 ;
hence not 41 divides 5923 by NAT_4:9; :: thesis:
5923 = (43 * 137) + 32 ;
hence not 43 divides 5923 by NAT_4:9; :: thesis:
5923 = (47 * 126) + 1 ;
hence not 47 divides 5923 by NAT_4:9; :: thesis:
5923 = (53 * 111) + 40 ;
hence not 53 divides 5923 by NAT_4:9; :: thesis:
5923 = (59 * 100) + 23 ;
hence not 59 divides 5923 by NAT_4:9; :: thesis:
5923 = (61 * 97) + 6 ;
hence not 61 divides 5923 by NAT_4:9; :: thesis:
5923 = (67 * 88) + 27 ;
hence not 67 divides 5923 by NAT_4:9; :: thesis:
5923 = (71 * 83) + 30 ;
hence not 71 divides 5923 by NAT_4:9; :: thesis:
5923 = (73 * 81) + 10 ;
hence not 73 divides 5923 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5923 & n is prime holds
not n divides 5923 by XPRIMET1:42;
hence 5923 is prime by NAT_4:14; :: thesis: