7069 = (2 * 3534) + 1 ;
hence not 2 divides 7069 by NAT_4:9; :: thesis:
7069 = (3 * 2356) + 1 ;
hence not 3 divides 7069 by NAT_4:9; :: thesis:
7069 = (5 * 1413) + 4 ;
hence not 5 divides 7069 by NAT_4:9; :: thesis:
7069 = (7 * 1009) + 6 ;
hence not 7 divides 7069 by NAT_4:9; :: thesis:
7069 = (11 * 642) + 7 ;
hence not 11 divides 7069 by NAT_4:9; :: thesis:
7069 = (13 * 543) + 10 ;
hence not 13 divides 7069 by NAT_4:9; :: thesis:
7069 = (17 * 415) + 14 ;
hence not 17 divides 7069 by NAT_4:9; :: thesis:
7069 = (19 * 372) + 1 ;
hence not 19 divides 7069 by NAT_4:9; :: thesis:
7069 = (23 * 307) + 8 ;
hence not 23 divides 7069 by NAT_4:9; :: thesis:
7069 = (29 * 243) + 22 ;
hence not 29 divides 7069 by NAT_4:9; :: thesis:
7069 = (31 * 228) + 1 ;
hence not 31 divides 7069 by NAT_4:9; :: thesis:
7069 = (37 * 191) + 2 ;
hence not 37 divides 7069 by NAT_4:9; :: thesis:
7069 = (41 * 172) + 17 ;
hence not 41 divides 7069 by NAT_4:9; :: thesis:
7069 = (43 * 164) + 17 ;
hence not 43 divides 7069 by NAT_4:9; :: thesis:
7069 = (47 * 150) + 19 ;
hence not 47 divides 7069 by NAT_4:9; :: thesis:
7069 = (53 * 133) + 20 ;
hence not 53 divides 7069 by NAT_4:9; :: thesis:
7069 = (59 * 119) + 48 ;
hence not 59 divides 7069 by NAT_4:9; :: thesis:
7069 = (61 * 115) + 54 ;
hence not 61 divides 7069 by NAT_4:9; :: thesis:
7069 = (67 * 105) + 34 ;
hence not 67 divides 7069 by NAT_4:9; :: thesis:
7069 = (71 * 99) + 40 ;
hence not 71 divides 7069 by NAT_4:9; :: thesis:
7069 = (73 * 96) + 61 ;
hence not 73 divides 7069 by NAT_4:9; :: thesis:
7069 = (79 * 89) + 38 ;
hence not 79 divides 7069 by NAT_4:9; :: thesis:
7069 = (83 * 85) + 14 ;
hence not 83 divides 7069 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7069 & n is prime holds
not n divides 7069 by XPRIMET1:46;
hence 7069 is prime by NAT_4:14; :: thesis: