7043 = (2 * 3521) + 1 ;
hence not 2 divides 7043 by NAT_4:9; :: thesis:
7043 = (3 * 2347) + 2 ;
hence not 3 divides 7043 by NAT_4:9; :: thesis:
7043 = (5 * 1408) + 3 ;
hence not 5 divides 7043 by NAT_4:9; :: thesis:
7043 = (7 * 1006) + 1 ;
hence not 7 divides 7043 by NAT_4:9; :: thesis:
7043 = (11 * 640) + 3 ;
hence not 11 divides 7043 by NAT_4:9; :: thesis:
7043 = (13 * 541) + 10 ;
hence not 13 divides 7043 by NAT_4:9; :: thesis:
7043 = (17 * 414) + 5 ;
hence not 17 divides 7043 by NAT_4:9; :: thesis:
7043 = (19 * 370) + 13 ;
hence not 19 divides 7043 by NAT_4:9; :: thesis:
7043 = (23 * 306) + 5 ;
hence not 23 divides 7043 by NAT_4:9; :: thesis:
7043 = (29 * 242) + 25 ;
hence not 29 divides 7043 by NAT_4:9; :: thesis:
7043 = (31 * 227) + 6 ;
hence not 31 divides 7043 by NAT_4:9; :: thesis:
7043 = (37 * 190) + 13 ;
hence not 37 divides 7043 by NAT_4:9; :: thesis:
7043 = (41 * 171) + 32 ;
hence not 41 divides 7043 by NAT_4:9; :: thesis:
7043 = (43 * 163) + 34 ;
hence not 43 divides 7043 by NAT_4:9; :: thesis:
7043 = (47 * 149) + 40 ;
hence not 47 divides 7043 by NAT_4:9; :: thesis:
7043 = (53 * 132) + 47 ;
hence not 53 divides 7043 by NAT_4:9; :: thesis:
7043 = (59 * 119) + 22 ;
hence not 59 divides 7043 by NAT_4:9; :: thesis:
7043 = (61 * 115) + 28 ;
hence not 61 divides 7043 by NAT_4:9; :: thesis:
7043 = (67 * 105) + 8 ;
hence not 67 divides 7043 by NAT_4:9; :: thesis:
7043 = (71 * 99) + 14 ;
hence not 71 divides 7043 by NAT_4:9; :: thesis:
7043 = (73 * 96) + 35 ;
hence not 73 divides 7043 by NAT_4:9; :: thesis:
7043 = (79 * 89) + 12 ;
hence not 79 divides 7043 by NAT_4:9; :: thesis:
7043 = (83 * 84) + 71 ;
hence not 83 divides 7043 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7043 & n is prime holds
not n divides 7043 by XPRIMET1:46;
hence 7043 is prime by NAT_4:14; :: thesis: