2953 = (2 * 1476) + 1 ;
hence not 2 divides 2953 by NAT_4:9; :: thesis:
2953 = (3 * 984) + 1 ;
hence not 3 divides 2953 by NAT_4:9; :: thesis:
2953 = (5 * 590) + 3 ;
hence not 5 divides 2953 by NAT_4:9; :: thesis:
2953 = (7 * 421) + 6 ;
hence not 7 divides 2953 by NAT_4:9; :: thesis:
2953 = (11 * 268) + 5 ;
hence not 11 divides 2953 by NAT_4:9; :: thesis:
2953 = (13 * 227) + 2 ;
hence not 13 divides 2953 by NAT_4:9; :: thesis:
2953 = (17 * 173) + 12 ;
hence not 17 divides 2953 by NAT_4:9; :: thesis:
2953 = (19 * 155) + 8 ;
hence not 19 divides 2953 by NAT_4:9; :: thesis:
2953 = (23 * 128) + 9 ;
hence not 23 divides 2953 by NAT_4:9; :: thesis:
2953 = (29 * 101) + 24 ;
hence not 29 divides 2953 by NAT_4:9; :: thesis:
2953 = (31 * 95) + 8 ;
hence not 31 divides 2953 by NAT_4:9; :: thesis:
2953 = (37 * 79) + 30 ;
hence not 37 divides 2953 by NAT_4:9; :: thesis:
2953 = (41 * 72) + 1 ;
hence not 41 divides 2953 by NAT_4:9; :: thesis:
2953 = (43 * 68) + 29 ;
hence not 43 divides 2953 by NAT_4:9; :: thesis:
2953 = (47 * 62) + 39 ;
hence not 47 divides 2953 by NAT_4:9; :: thesis:
2953 = (53 * 55) + 38 ;
hence not 53 divides 2953 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2953 & n is prime holds
not n divides 2953 by XPRIMET1:32;
hence 2953 is prime by NAT_4:14; :: thesis: