4217 = (2 * 2108) + 1 ;
hence not 2 divides 4217 by NAT_4:9; :: thesis:
4217 = (3 * 1405) + 2 ;
hence not 3 divides 4217 by NAT_4:9; :: thesis:
4217 = (5 * 843) + 2 ;
hence not 5 divides 4217 by NAT_4:9; :: thesis:
4217 = (7 * 602) + 3 ;
hence not 7 divides 4217 by NAT_4:9; :: thesis:
4217 = (11 * 383) + 4 ;
hence not 11 divides 4217 by NAT_4:9; :: thesis:
4217 = (13 * 324) + 5 ;
hence not 13 divides 4217 by NAT_4:9; :: thesis:
4217 = (17 * 248) + 1 ;
hence not 17 divides 4217 by NAT_4:9; :: thesis:
4217 = (19 * 221) + 18 ;
hence not 19 divides 4217 by NAT_4:9; :: thesis:
4217 = (23 * 183) + 8 ;
hence not 23 divides 4217 by NAT_4:9; :: thesis:
4217 = (29 * 145) + 12 ;
hence not 29 divides 4217 by NAT_4:9; :: thesis:
4217 = (31 * 136) + 1 ;
hence not 31 divides 4217 by NAT_4:9; :: thesis:
4217 = (37 * 113) + 36 ;
hence not 37 divides 4217 by NAT_4:9; :: thesis:
4217 = (41 * 102) + 35 ;
hence not 41 divides 4217 by NAT_4:9; :: thesis:
4217 = (43 * 98) + 3 ;
hence not 43 divides 4217 by NAT_4:9; :: thesis:
4217 = (47 * 89) + 34 ;
hence not 47 divides 4217 by NAT_4:9; :: thesis:
4217 = (53 * 79) + 30 ;
hence not 53 divides 4217 by NAT_4:9; :: thesis:
4217 = (59 * 71) + 28 ;
hence not 59 divides 4217 by NAT_4:9; :: thesis:
4217 = (61 * 69) + 8 ;
hence not 61 divides 4217 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4217 & n is prime holds
not n divides 4217 by XPRIMET1:36;
hence 4217 is prime by NAT_4:14; :: thesis: