7039 = (2 * 3519) + 1 ;
hence not 2 divides 7039 by NAT_4:9; :: thesis:
7039 = (3 * 2346) + 1 ;
hence not 3 divides 7039 by NAT_4:9; :: thesis:
7039 = (5 * 1407) + 4 ;
hence not 5 divides 7039 by NAT_4:9; :: thesis:
7039 = (7 * 1005) + 4 ;
hence not 7 divides 7039 by NAT_4:9; :: thesis:
7039 = (11 * 639) + 10 ;
hence not 11 divides 7039 by NAT_4:9; :: thesis:
7039 = (13 * 541) + 6 ;
hence not 13 divides 7039 by NAT_4:9; :: thesis:
7039 = (17 * 414) + 1 ;
hence not 17 divides 7039 by NAT_4:9; :: thesis:
7039 = (19 * 370) + 9 ;
hence not 19 divides 7039 by NAT_4:9; :: thesis:
7039 = (23 * 306) + 1 ;
hence not 23 divides 7039 by NAT_4:9; :: thesis:
7039 = (29 * 242) + 21 ;
hence not 29 divides 7039 by NAT_4:9; :: thesis:
7039 = (31 * 227) + 2 ;
hence not 31 divides 7039 by NAT_4:9; :: thesis:
7039 = (37 * 190) + 9 ;
hence not 37 divides 7039 by NAT_4:9; :: thesis:
7039 = (41 * 171) + 28 ;
hence not 41 divides 7039 by NAT_4:9; :: thesis:
7039 = (43 * 163) + 30 ;
hence not 43 divides 7039 by NAT_4:9; :: thesis:
7039 = (47 * 149) + 36 ;
hence not 47 divides 7039 by NAT_4:9; :: thesis:
7039 = (53 * 132) + 43 ;
hence not 53 divides 7039 by NAT_4:9; :: thesis:
7039 = (59 * 119) + 18 ;
hence not 59 divides 7039 by NAT_4:9; :: thesis:
7039 = (61 * 115) + 24 ;
hence not 61 divides 7039 by NAT_4:9; :: thesis:
7039 = (67 * 105) + 4 ;
hence not 67 divides 7039 by NAT_4:9; :: thesis:
7039 = (71 * 99) + 10 ;
hence not 71 divides 7039 by NAT_4:9; :: thesis:
7039 = (73 * 96) + 31 ;
hence not 73 divides 7039 by NAT_4:9; :: thesis:
7039 = (79 * 89) + 8 ;
hence not 79 divides 7039 by NAT_4:9; :: thesis:
7039 = (83 * 84) + 67 ;
hence not 83 divides 7039 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7039 & n is prime holds
not n divides 7039 by XPRIMET1:46;
hence 7039 is prime by NAT_4:14; :: thesis: