4507 = (2 * 2253) + 1 ;
hence not 2 divides 4507 by NAT_4:9; :: thesis:
4507 = (3 * 1502) + 1 ;
hence not 3 divides 4507 by NAT_4:9; :: thesis:
4507 = (5 * 901) + 2 ;
hence not 5 divides 4507 by NAT_4:9; :: thesis:
4507 = (7 * 643) + 6 ;
hence not 7 divides 4507 by NAT_4:9; :: thesis:
4507 = (11 * 409) + 8 ;
hence not 11 divides 4507 by NAT_4:9; :: thesis:
4507 = (13 * 346) + 9 ;
hence not 13 divides 4507 by NAT_4:9; :: thesis:
4507 = (17 * 265) + 2 ;
hence not 17 divides 4507 by NAT_4:9; :: thesis:
4507 = (19 * 237) + 4 ;
hence not 19 divides 4507 by NAT_4:9; :: thesis:
4507 = (23 * 195) + 22 ;
hence not 23 divides 4507 by NAT_4:9; :: thesis:
4507 = (29 * 155) + 12 ;
hence not 29 divides 4507 by NAT_4:9; :: thesis:
4507 = (31 * 145) + 12 ;
hence not 31 divides 4507 by NAT_4:9; :: thesis:
4507 = (37 * 121) + 30 ;
hence not 37 divides 4507 by NAT_4:9; :: thesis:
4507 = (41 * 109) + 38 ;
hence not 41 divides 4507 by NAT_4:9; :: thesis:
4507 = (43 * 104) + 35 ;
hence not 43 divides 4507 by NAT_4:9; :: thesis:
4507 = (47 * 95) + 42 ;
hence not 47 divides 4507 by NAT_4:9; :: thesis:
4507 = (53 * 85) + 2 ;
hence not 53 divides 4507 by NAT_4:9; :: thesis:
4507 = (59 * 76) + 23 ;
hence not 59 divides 4507 by NAT_4:9; :: thesis:
4507 = (61 * 73) + 54 ;
hence not 61 divides 4507 by NAT_4:9; :: thesis:
4507 = (67 * 67) + 18 ;
hence not 67 divides 4507 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4507 & n is prime holds
not n divides 4507 by XPRIMET1:38;
hence 4507 is prime by NAT_4:14; :: thesis: