7823 = (2 * 3911) + 1 ;
hence not 2 divides 7823 by NAT_4:9; :: thesis:
7823 = (3 * 2607) + 2 ;
hence not 3 divides 7823 by NAT_4:9; :: thesis:
7823 = (5 * 1564) + 3 ;
hence not 5 divides 7823 by NAT_4:9; :: thesis:
7823 = (7 * 1117) + 4 ;
hence not 7 divides 7823 by NAT_4:9; :: thesis:
7823 = (11 * 711) + 2 ;
hence not 11 divides 7823 by NAT_4:9; :: thesis:
7823 = (13 * 601) + 10 ;
hence not 13 divides 7823 by NAT_4:9; :: thesis:
7823 = (17 * 460) + 3 ;
hence not 17 divides 7823 by NAT_4:9; :: thesis:
7823 = (19 * 411) + 14 ;
hence not 19 divides 7823 by NAT_4:9; :: thesis:
7823 = (23 * 340) + 3 ;
hence not 23 divides 7823 by NAT_4:9; :: thesis:
7823 = (29 * 269) + 22 ;
hence not 29 divides 7823 by NAT_4:9; :: thesis:
7823 = (31 * 252) + 11 ;
hence not 31 divides 7823 by NAT_4:9; :: thesis:
7823 = (37 * 211) + 16 ;
hence not 37 divides 7823 by NAT_4:9; :: thesis:
7823 = (41 * 190) + 33 ;
hence not 41 divides 7823 by NAT_4:9; :: thesis:
7823 = (43 * 181) + 40 ;
hence not 43 divides 7823 by NAT_4:9; :: thesis:
7823 = (47 * 166) + 21 ;
hence not 47 divides 7823 by NAT_4:9; :: thesis:
7823 = (53 * 147) + 32 ;
hence not 53 divides 7823 by NAT_4:9; :: thesis:
7823 = (59 * 132) + 35 ;
hence not 59 divides 7823 by NAT_4:9; :: thesis:
7823 = (61 * 128) + 15 ;
hence not 61 divides 7823 by NAT_4:9; :: thesis:
7823 = (67 * 116) + 51 ;
hence not 67 divides 7823 by NAT_4:9; :: thesis:
7823 = (71 * 110) + 13 ;
hence not 71 divides 7823 by NAT_4:9; :: thesis:
7823 = (73 * 107) + 12 ;
hence not 73 divides 7823 by NAT_4:9; :: thesis:
7823 = (79 * 99) + 2 ;
hence not 79 divides 7823 by NAT_4:9; :: thesis:
7823 = (83 * 94) + 21 ;
hence not 83 divides 7823 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7823 & n is prime holds
not n divides 7823 by XPRIMET1:46;
hence 7823 is prime by NAT_4:14; :: thesis: