4027 = (2 * 2013) + 1 ;
hence not 2 divides 4027 by NAT_4:9; :: thesis:
4027 = (3 * 1342) + 1 ;
hence not 3 divides 4027 by NAT_4:9; :: thesis:
4027 = (5 * 805) + 2 ;
hence not 5 divides 4027 by NAT_4:9; :: thesis:
4027 = (7 * 575) + 2 ;
hence not 7 divides 4027 by NAT_4:9; :: thesis:
4027 = (11 * 366) + 1 ;
hence not 11 divides 4027 by NAT_4:9; :: thesis:
4027 = (13 * 309) + 10 ;
hence not 13 divides 4027 by NAT_4:9; :: thesis:
4027 = (17 * 236) + 15 ;
hence not 17 divides 4027 by NAT_4:9; :: thesis:
4027 = (19 * 211) + 18 ;
hence not 19 divides 4027 by NAT_4:9; :: thesis:
4027 = (23 * 175) + 2 ;
hence not 23 divides 4027 by NAT_4:9; :: thesis:
4027 = (29 * 138) + 25 ;
hence not 29 divides 4027 by NAT_4:9; :: thesis:
4027 = (31 * 129) + 28 ;
hence not 31 divides 4027 by NAT_4:9; :: thesis:
4027 = (37 * 108) + 31 ;
hence not 37 divides 4027 by NAT_4:9; :: thesis:
4027 = (41 * 98) + 9 ;
hence not 41 divides 4027 by NAT_4:9; :: thesis:
4027 = (43 * 93) + 28 ;
hence not 43 divides 4027 by NAT_4:9; :: thesis:
4027 = (47 * 85) + 32 ;
hence not 47 divides 4027 by NAT_4:9; :: thesis:
4027 = (53 * 75) + 52 ;
hence not 53 divides 4027 by NAT_4:9; :: thesis:
4027 = (59 * 68) + 15 ;
hence not 59 divides 4027 by NAT_4:9; :: thesis:
4027 = (61 * 66) + 1 ;
hence not 61 divides 4027 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4027 & n is prime holds
not n divides 4027 by XPRIMET1:36;
hence 4027 is prime by NAT_4:14; :: thesis: