2027 = (2 * 1013) + 1 ;
hence not 2 divides 2027 by NAT_4:9; :: thesis:
2027 = (3 * 675) + 2 ;
hence not 3 divides 2027 by NAT_4:9; :: thesis:
2027 = (5 * 405) + 2 ;
hence not 5 divides 2027 by NAT_4:9; :: thesis:
2027 = (7 * 289) + 4 ;
hence not 7 divides 2027 by NAT_4:9; :: thesis:
2027 = (11 * 184) + 3 ;
hence not 11 divides 2027 by NAT_4:9; :: thesis:
2027 = (13 * 155) + 12 ;
hence not 13 divides 2027 by NAT_4:9; :: thesis:
2027 = (17 * 119) + 4 ;
hence not 17 divides 2027 by NAT_4:9; :: thesis:
2027 = (19 * 106) + 13 ;
hence not 19 divides 2027 by NAT_4:9; :: thesis:
2027 = (23 * 88) + 3 ;
hence not 23 divides 2027 by NAT_4:9; :: thesis:
2027 = (29 * 69) + 26 ;
hence not 29 divides 2027 by NAT_4:9; :: thesis:
2027 = (31 * 65) + 12 ;
hence not 31 divides 2027 by NAT_4:9; :: thesis:
2027 = (37 * 54) + 29 ;
hence not 37 divides 2027 by NAT_4:9; :: thesis:
2027 = (41 * 49) + 18 ;
hence not 41 divides 2027 by NAT_4:9; :: thesis:
2027 = (43 * 47) + 6 ;
hence not 43 divides 2027 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2027 & n is prime holds
not n divides 2027 by XPRIMET1:28;
hence 2027 is prime by NAT_4:14; :: thesis: