7643 = (2 * 3821) + 1 ;
hence not 2 divides 7643 by NAT_4:9; :: thesis:
7643 = (3 * 2547) + 2 ;
hence not 3 divides 7643 by NAT_4:9; :: thesis:
7643 = (5 * 1528) + 3 ;
hence not 5 divides 7643 by NAT_4:9; :: thesis:
7643 = (7 * 1091) + 6 ;
hence not 7 divides 7643 by NAT_4:9; :: thesis:
7643 = (11 * 694) + 9 ;
hence not 11 divides 7643 by NAT_4:9; :: thesis:
7643 = (13 * 587) + 12 ;
hence not 13 divides 7643 by NAT_4:9; :: thesis:
7643 = (17 * 449) + 10 ;
hence not 17 divides 7643 by NAT_4:9; :: thesis:
7643 = (19 * 402) + 5 ;
hence not 19 divides 7643 by NAT_4:9; :: thesis:
7643 = (23 * 332) + 7 ;
hence not 23 divides 7643 by NAT_4:9; :: thesis:
7643 = (29 * 263) + 16 ;
hence not 29 divides 7643 by NAT_4:9; :: thesis:
7643 = (31 * 246) + 17 ;
hence not 31 divides 7643 by NAT_4:9; :: thesis:
7643 = (37 * 206) + 21 ;
hence not 37 divides 7643 by NAT_4:9; :: thesis:
7643 = (41 * 186) + 17 ;
hence not 41 divides 7643 by NAT_4:9; :: thesis:
7643 = (43 * 177) + 32 ;
hence not 43 divides 7643 by NAT_4:9; :: thesis:
7643 = (47 * 162) + 29 ;
hence not 47 divides 7643 by NAT_4:9; :: thesis:
7643 = (53 * 144) + 11 ;
hence not 53 divides 7643 by NAT_4:9; :: thesis:
7643 = (59 * 129) + 32 ;
hence not 59 divides 7643 by NAT_4:9; :: thesis:
7643 = (61 * 125) + 18 ;
hence not 61 divides 7643 by NAT_4:9; :: thesis:
7643 = (67 * 114) + 5 ;
hence not 67 divides 7643 by NAT_4:9; :: thesis:
7643 = (71 * 107) + 46 ;
hence not 71 divides 7643 by NAT_4:9; :: thesis:
7643 = (73 * 104) + 51 ;
hence not 73 divides 7643 by NAT_4:9; :: thesis:
7643 = (79 * 96) + 59 ;
hence not 79 divides 7643 by NAT_4:9; :: thesis:
7643 = (83 * 92) + 7 ;
hence not 83 divides 7643 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7643 & n is prime holds
not n divides 7643 by XPRIMET1:46;
hence 7643 is prime by NAT_4:14; :: thesis: