7853 = (2 * 3926) + 1 ;
hence not 2 divides 7853 by NAT_4:9; :: thesis:
7853 = (3 * 2617) + 2 ;
hence not 3 divides 7853 by NAT_4:9; :: thesis:
7853 = (5 * 1570) + 3 ;
hence not 5 divides 7853 by NAT_4:9; :: thesis:
7853 = (7 * 1121) + 6 ;
hence not 7 divides 7853 by NAT_4:9; :: thesis:
7853 = (11 * 713) + 10 ;
hence not 11 divides 7853 by NAT_4:9; :: thesis:
7853 = (13 * 604) + 1 ;
hence not 13 divides 7853 by NAT_4:9; :: thesis:
7853 = (17 * 461) + 16 ;
hence not 17 divides 7853 by NAT_4:9; :: thesis:
7853 = (19 * 413) + 6 ;
hence not 19 divides 7853 by NAT_4:9; :: thesis:
7853 = (23 * 341) + 10 ;
hence not 23 divides 7853 by NAT_4:9; :: thesis:
7853 = (29 * 270) + 23 ;
hence not 29 divides 7853 by NAT_4:9; :: thesis:
7853 = (31 * 253) + 10 ;
hence not 31 divides 7853 by NAT_4:9; :: thesis:
7853 = (37 * 212) + 9 ;
hence not 37 divides 7853 by NAT_4:9; :: thesis:
7853 = (41 * 191) + 22 ;
hence not 41 divides 7853 by NAT_4:9; :: thesis:
7853 = (43 * 182) + 27 ;
hence not 43 divides 7853 by NAT_4:9; :: thesis:
7853 = (47 * 167) + 4 ;
hence not 47 divides 7853 by NAT_4:9; :: thesis:
7853 = (53 * 148) + 9 ;
hence not 53 divides 7853 by NAT_4:9; :: thesis:
7853 = (59 * 133) + 6 ;
hence not 59 divides 7853 by NAT_4:9; :: thesis:
7853 = (61 * 128) + 45 ;
hence not 61 divides 7853 by NAT_4:9; :: thesis:
7853 = (67 * 117) + 14 ;
hence not 67 divides 7853 by NAT_4:9; :: thesis:
7853 = (71 * 110) + 43 ;
hence not 71 divides 7853 by NAT_4:9; :: thesis:
7853 = (73 * 107) + 42 ;
hence not 73 divides 7853 by NAT_4:9; :: thesis:
7853 = (79 * 99) + 32 ;
hence not 79 divides 7853 by NAT_4:9; :: thesis:
7853 = (83 * 94) + 51 ;
hence not 83 divides 7853 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7853 & n is prime holds
not n divides 7853 by XPRIMET1:46;
hence 7853 is prime by NAT_4:14; :: thesis: