7639 = (2 * 3819) + 1 ;
hence not 2 divides 7639 by NAT_4:9; :: thesis:
7639 = (3 * 2546) + 1 ;
hence not 3 divides 7639 by NAT_4:9; :: thesis:
7639 = (5 * 1527) + 4 ;
hence not 5 divides 7639 by NAT_4:9; :: thesis:
7639 = (7 * 1091) + 2 ;
hence not 7 divides 7639 by NAT_4:9; :: thesis:
7639 = (11 * 694) + 5 ;
hence not 11 divides 7639 by NAT_4:9; :: thesis:
7639 = (13 * 587) + 8 ;
hence not 13 divides 7639 by NAT_4:9; :: thesis:
7639 = (17 * 449) + 6 ;
hence not 17 divides 7639 by NAT_4:9; :: thesis:
7639 = (19 * 402) + 1 ;
hence not 19 divides 7639 by NAT_4:9; :: thesis:
7639 = (23 * 332) + 3 ;
hence not 23 divides 7639 by NAT_4:9; :: thesis:
7639 = (29 * 263) + 12 ;
hence not 29 divides 7639 by NAT_4:9; :: thesis:
7639 = (31 * 246) + 13 ;
hence not 31 divides 7639 by NAT_4:9; :: thesis:
7639 = (37 * 206) + 17 ;
hence not 37 divides 7639 by NAT_4:9; :: thesis:
7639 = (41 * 186) + 13 ;
hence not 41 divides 7639 by NAT_4:9; :: thesis:
7639 = (43 * 177) + 28 ;
hence not 43 divides 7639 by NAT_4:9; :: thesis:
7639 = (47 * 162) + 25 ;
hence not 47 divides 7639 by NAT_4:9; :: thesis:
7639 = (53 * 144) + 7 ;
hence not 53 divides 7639 by NAT_4:9; :: thesis:
7639 = (59 * 129) + 28 ;
hence not 59 divides 7639 by NAT_4:9; :: thesis:
7639 = (61 * 125) + 14 ;
hence not 61 divides 7639 by NAT_4:9; :: thesis:
7639 = (67 * 114) + 1 ;
hence not 67 divides 7639 by NAT_4:9; :: thesis:
7639 = (71 * 107) + 42 ;
hence not 71 divides 7639 by NAT_4:9; :: thesis:
7639 = (73 * 104) + 47 ;
hence not 73 divides 7639 by NAT_4:9; :: thesis:
7639 = (79 * 96) + 55 ;
hence not 79 divides 7639 by NAT_4:9; :: thesis:
7639 = (83 * 92) + 3 ;
hence not 83 divides 7639 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7639 & n is prime holds
not n divides 7639 by XPRIMET1:46;
hence 7639 is prime by NAT_4:14; :: thesis: