5227 = (2 * 2613) + 1 ;
hence not 2 divides 5227 by NAT_4:9; :: thesis:
5227 = (3 * 1742) + 1 ;
hence not 3 divides 5227 by NAT_4:9; :: thesis:
5227 = (5 * 1045) + 2 ;
hence not 5 divides 5227 by NAT_4:9; :: thesis:
5227 = (7 * 746) + 5 ;
hence not 7 divides 5227 by NAT_4:9; :: thesis:
5227 = (11 * 475) + 2 ;
hence not 11 divides 5227 by NAT_4:9; :: thesis:
5227 = (13 * 402) + 1 ;
hence not 13 divides 5227 by NAT_4:9; :: thesis:
5227 = (17 * 307) + 8 ;
hence not 17 divides 5227 by NAT_4:9; :: thesis:
5227 = (19 * 275) + 2 ;
hence not 19 divides 5227 by NAT_4:9; :: thesis:
5227 = (23 * 227) + 6 ;
hence not 23 divides 5227 by NAT_4:9; :: thesis:
5227 = (29 * 180) + 7 ;
hence not 29 divides 5227 by NAT_4:9; :: thesis:
5227 = (31 * 168) + 19 ;
hence not 31 divides 5227 by NAT_4:9; :: thesis:
5227 = (37 * 141) + 10 ;
hence not 37 divides 5227 by NAT_4:9; :: thesis:
5227 = (41 * 127) + 20 ;
hence not 41 divides 5227 by NAT_4:9; :: thesis:
5227 = (43 * 121) + 24 ;
hence not 43 divides 5227 by NAT_4:9; :: thesis:
5227 = (47 * 111) + 10 ;
hence not 47 divides 5227 by NAT_4:9; :: thesis:
5227 = (53 * 98) + 33 ;
hence not 53 divides 5227 by NAT_4:9; :: thesis:
5227 = (59 * 88) + 35 ;
hence not 59 divides 5227 by NAT_4:9; :: thesis:
5227 = (61 * 85) + 42 ;
hence not 61 divides 5227 by NAT_4:9; :: thesis:
5227 = (67 * 78) + 1 ;
hence not 67 divides 5227 by NAT_4:9; :: thesis:
5227 = (71 * 73) + 44 ;
hence not 71 divides 5227 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5227 & n is prime holds
not n divides 5227 by XPRIMET1:40;
hence 5227 is prime by NAT_4:14; :: thesis: