4729 = (2 * 2364) + 1 ;
hence not 2 divides 4729 by NAT_4:9; :: thesis:
4729 = (3 * 1576) + 1 ;
hence not 3 divides 4729 by NAT_4:9; :: thesis:
4729 = (5 * 945) + 4 ;
hence not 5 divides 4729 by NAT_4:9; :: thesis:
4729 = (7 * 675) + 4 ;
hence not 7 divides 4729 by NAT_4:9; :: thesis:
4729 = (11 * 429) + 10 ;
hence not 11 divides 4729 by NAT_4:9; :: thesis:
4729 = (13 * 363) + 10 ;
hence not 13 divides 4729 by NAT_4:9; :: thesis:
4729 = (17 * 278) + 3 ;
hence not 17 divides 4729 by NAT_4:9; :: thesis:
4729 = (19 * 248) + 17 ;
hence not 19 divides 4729 by NAT_4:9; :: thesis:
4729 = (23 * 205) + 14 ;
hence not 23 divides 4729 by NAT_4:9; :: thesis:
4729 = (29 * 163) + 2 ;
hence not 29 divides 4729 by NAT_4:9; :: thesis:
4729 = (31 * 152) + 17 ;
hence not 31 divides 4729 by NAT_4:9; :: thesis:
4729 = (37 * 127) + 30 ;
hence not 37 divides 4729 by NAT_4:9; :: thesis:
4729 = (41 * 115) + 14 ;
hence not 41 divides 4729 by NAT_4:9; :: thesis:
4729 = (43 * 109) + 42 ;
hence not 43 divides 4729 by NAT_4:9; :: thesis:
4729 = (47 * 100) + 29 ;
hence not 47 divides 4729 by NAT_4:9; :: thesis:
4729 = (53 * 89) + 12 ;
hence not 53 divides 4729 by NAT_4:9; :: thesis:
4729 = (59 * 80) + 9 ;
hence not 59 divides 4729 by NAT_4:9; :: thesis:
4729 = (61 * 77) + 32 ;
hence not 61 divides 4729 by NAT_4:9; :: thesis:
4729 = (67 * 70) + 39 ;
hence not 67 divides 4729 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4729 & n is prime holds
not n divides 4729 by XPRIMET1:38;
hence 4729 is prime by NAT_4:14; :: thesis: