7621 = (2 * 3810) + 1 ;
hence not 2 divides 7621 by NAT_4:9; :: thesis:
7621 = (3 * 2540) + 1 ;
hence not 3 divides 7621 by NAT_4:9; :: thesis:
7621 = (5 * 1524) + 1 ;
hence not 5 divides 7621 by NAT_4:9; :: thesis:
7621 = (7 * 1088) + 5 ;
hence not 7 divides 7621 by NAT_4:9; :: thesis:
7621 = (11 * 692) + 9 ;
hence not 11 divides 7621 by NAT_4:9; :: thesis:
7621 = (13 * 586) + 3 ;
hence not 13 divides 7621 by NAT_4:9; :: thesis:
7621 = (17 * 448) + 5 ;
hence not 17 divides 7621 by NAT_4:9; :: thesis:
7621 = (19 * 401) + 2 ;
hence not 19 divides 7621 by NAT_4:9; :: thesis:
7621 = (23 * 331) + 8 ;
hence not 23 divides 7621 by NAT_4:9; :: thesis:
7621 = (29 * 262) + 23 ;
hence not 29 divides 7621 by NAT_4:9; :: thesis:
7621 = (31 * 245) + 26 ;
hence not 31 divides 7621 by NAT_4:9; :: thesis:
7621 = (37 * 205) + 36 ;
hence not 37 divides 7621 by NAT_4:9; :: thesis:
7621 = (41 * 185) + 36 ;
hence not 41 divides 7621 by NAT_4:9; :: thesis:
7621 = (43 * 177) + 10 ;
hence not 43 divides 7621 by NAT_4:9; :: thesis:
7621 = (47 * 162) + 7 ;
hence not 47 divides 7621 by NAT_4:9; :: thesis:
7621 = (53 * 143) + 42 ;
hence not 53 divides 7621 by NAT_4:9; :: thesis:
7621 = (59 * 129) + 10 ;
hence not 59 divides 7621 by NAT_4:9; :: thesis:
7621 = (61 * 124) + 57 ;
hence not 61 divides 7621 by NAT_4:9; :: thesis:
7621 = (67 * 113) + 50 ;
hence not 67 divides 7621 by NAT_4:9; :: thesis:
7621 = (71 * 107) + 24 ;
hence not 71 divides 7621 by NAT_4:9; :: thesis:
7621 = (73 * 104) + 29 ;
hence not 73 divides 7621 by NAT_4:9; :: thesis:
7621 = (79 * 96) + 37 ;
hence not 79 divides 7621 by NAT_4:9; :: thesis:
7621 = (83 * 91) + 68 ;
hence not 83 divides 7621 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7621 & n is prime holds
not n divides 7621 by XPRIMET1:46;
hence 7621 is prime by NAT_4:14; :: thesis: