4201 = (2 * 2100) + 1 ;
hence not 2 divides 4201 by NAT_4:9; :: thesis:
4201 = (3 * 1400) + 1 ;
hence not 3 divides 4201 by NAT_4:9; :: thesis:
4201 = (5 * 840) + 1 ;
hence not 5 divides 4201 by NAT_4:9; :: thesis:
4201 = (7 * 600) + 1 ;
hence not 7 divides 4201 by NAT_4:9; :: thesis:
4201 = (11 * 381) + 10 ;
hence not 11 divides 4201 by NAT_4:9; :: thesis:
4201 = (13 * 323) + 2 ;
hence not 13 divides 4201 by NAT_4:9; :: thesis:
4201 = (17 * 247) + 2 ;
hence not 17 divides 4201 by NAT_4:9; :: thesis:
4201 = (19 * 221) + 2 ;
hence not 19 divides 4201 by NAT_4:9; :: thesis:
4201 = (23 * 182) + 15 ;
hence not 23 divides 4201 by NAT_4:9; :: thesis:
4201 = (29 * 144) + 25 ;
hence not 29 divides 4201 by NAT_4:9; :: thesis:
4201 = (31 * 135) + 16 ;
hence not 31 divides 4201 by NAT_4:9; :: thesis:
4201 = (37 * 113) + 20 ;
hence not 37 divides 4201 by NAT_4:9; :: thesis:
4201 = (41 * 102) + 19 ;
hence not 41 divides 4201 by NAT_4:9; :: thesis:
4201 = (43 * 97) + 30 ;
hence not 43 divides 4201 by NAT_4:9; :: thesis:
4201 = (47 * 89) + 18 ;
hence not 47 divides 4201 by NAT_4:9; :: thesis:
4201 = (53 * 79) + 14 ;
hence not 53 divides 4201 by NAT_4:9; :: thesis:
4201 = (59 * 71) + 12 ;
hence not 59 divides 4201 by NAT_4:9; :: thesis:
4201 = (61 * 68) + 53 ;
hence not 61 divides 4201 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4201 & n is prime holds
not n divides 4201 by XPRIMET1:36;
hence 4201 is prime by NAT_4:14; :: thesis: