6763 = (2 * 3381) + 1 ;
hence not 2 divides 6763 by NAT_4:9; :: thesis:
6763 = (3 * 2254) + 1 ;
hence not 3 divides 6763 by NAT_4:9; :: thesis:
6763 = (5 * 1352) + 3 ;
hence not 5 divides 6763 by NAT_4:9; :: thesis:
6763 = (7 * 966) + 1 ;
hence not 7 divides 6763 by NAT_4:9; :: thesis:
6763 = (11 * 614) + 9 ;
hence not 11 divides 6763 by NAT_4:9; :: thesis:
6763 = (13 * 520) + 3 ;
hence not 13 divides 6763 by NAT_4:9; :: thesis:
6763 = (17 * 397) + 14 ;
hence not 17 divides 6763 by NAT_4:9; :: thesis:
6763 = (19 * 355) + 18 ;
hence not 19 divides 6763 by NAT_4:9; :: thesis:
6763 = (23 * 294) + 1 ;
hence not 23 divides 6763 by NAT_4:9; :: thesis:
6763 = (29 * 233) + 6 ;
hence not 29 divides 6763 by NAT_4:9; :: thesis:
6763 = (31 * 218) + 5 ;
hence not 31 divides 6763 by NAT_4:9; :: thesis:
6763 = (37 * 182) + 29 ;
hence not 37 divides 6763 by NAT_4:9; :: thesis:
6763 = (41 * 164) + 39 ;
hence not 41 divides 6763 by NAT_4:9; :: thesis:
6763 = (43 * 157) + 12 ;
hence not 43 divides 6763 by NAT_4:9; :: thesis:
6763 = (47 * 143) + 42 ;
hence not 47 divides 6763 by NAT_4:9; :: thesis:
6763 = (53 * 127) + 32 ;
hence not 53 divides 6763 by NAT_4:9; :: thesis:
6763 = (59 * 114) + 37 ;
hence not 59 divides 6763 by NAT_4:9; :: thesis:
6763 = (61 * 110) + 53 ;
hence not 61 divides 6763 by NAT_4:9; :: thesis:
6763 = (67 * 100) + 63 ;
hence not 67 divides 6763 by NAT_4:9; :: thesis:
6763 = (71 * 95) + 18 ;
hence not 71 divides 6763 by NAT_4:9; :: thesis:
6763 = (73 * 92) + 47 ;
hence not 73 divides 6763 by NAT_4:9; :: thesis:
6763 = (79 * 85) + 48 ;
hence not 79 divides 6763 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6763 & n is prime holds
not n divides 6763 by XPRIMET1:44;
hence 6763 is prime by NAT_4:14; :: thesis: