6653 = (2 * 3326) + 1 ;
hence not 2 divides 6653 by NAT_4:9; :: thesis:
6653 = (3 * 2217) + 2 ;
hence not 3 divides 6653 by NAT_4:9; :: thesis:
6653 = (5 * 1330) + 3 ;
hence not 5 divides 6653 by NAT_4:9; :: thesis:
6653 = (7 * 950) + 3 ;
hence not 7 divides 6653 by NAT_4:9; :: thesis:
6653 = (11 * 604) + 9 ;
hence not 11 divides 6653 by NAT_4:9; :: thesis:
6653 = (13 * 511) + 10 ;
hence not 13 divides 6653 by NAT_4:9; :: thesis:
6653 = (17 * 391) + 6 ;
hence not 17 divides 6653 by NAT_4:9; :: thesis:
6653 = (19 * 350) + 3 ;
hence not 19 divides 6653 by NAT_4:9; :: thesis:
6653 = (23 * 289) + 6 ;
hence not 23 divides 6653 by NAT_4:9; :: thesis:
6653 = (29 * 229) + 12 ;
hence not 29 divides 6653 by NAT_4:9; :: thesis:
6653 = (31 * 214) + 19 ;
hence not 31 divides 6653 by NAT_4:9; :: thesis:
6653 = (37 * 179) + 30 ;
hence not 37 divides 6653 by NAT_4:9; :: thesis:
6653 = (41 * 162) + 11 ;
hence not 41 divides 6653 by NAT_4:9; :: thesis:
6653 = (43 * 154) + 31 ;
hence not 43 divides 6653 by NAT_4:9; :: thesis:
6653 = (47 * 141) + 26 ;
hence not 47 divides 6653 by NAT_4:9; :: thesis:
6653 = (53 * 125) + 28 ;
hence not 53 divides 6653 by NAT_4:9; :: thesis:
6653 = (59 * 112) + 45 ;
hence not 59 divides 6653 by NAT_4:9; :: thesis:
6653 = (61 * 109) + 4 ;
hence not 61 divides 6653 by NAT_4:9; :: thesis:
6653 = (67 * 99) + 20 ;
hence not 67 divides 6653 by NAT_4:9; :: thesis:
6653 = (71 * 93) + 50 ;
hence not 71 divides 6653 by NAT_4:9; :: thesis:
6653 = (73 * 91) + 10 ;
hence not 73 divides 6653 by NAT_4:9; :: thesis:
6653 = (79 * 84) + 17 ;
hence not 79 divides 6653 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6653 & n is prime holds
not n divides 6653 by XPRIMET1:44;
hence 6653 is prime by NAT_4:14; :: thesis: