2011 = (2 * 1005) + 1 ;
hence not 2 divides 2011 by NAT_4:9; :: thesis:
2011 = (3 * 670) + 1 ;
hence not 3 divides 2011 by NAT_4:9; :: thesis:
2011 = (5 * 402) + 1 ;
hence not 5 divides 2011 by NAT_4:9; :: thesis:
2011 = (7 * 287) + 2 ;
hence not 7 divides 2011 by NAT_4:9; :: thesis:
2011 = (11 * 182) + 9 ;
hence not 11 divides 2011 by NAT_4:9; :: thesis:
2011 = (13 * 154) + 9 ;
hence not 13 divides 2011 by NAT_4:9; :: thesis:
2011 = (17 * 118) + 5 ;
hence not 17 divides 2011 by NAT_4:9; :: thesis:
2011 = (19 * 105) + 16 ;
hence not 19 divides 2011 by NAT_4:9; :: thesis:
2011 = (23 * 87) + 10 ;
hence not 23 divides 2011 by NAT_4:9; :: thesis:
2011 = (29 * 69) + 10 ;
hence not 29 divides 2011 by NAT_4:9; :: thesis:
2011 = (31 * 64) + 27 ;
hence not 31 divides 2011 by NAT_4:9; :: thesis:
2011 = (37 * 54) + 13 ;
hence not 37 divides 2011 by NAT_4:9; :: thesis:
2011 = (41 * 49) + 2 ;
hence not 41 divides 2011 by NAT_4:9; :: thesis:
2011 = (43 * 46) + 33 ;
hence not 43 divides 2011 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2011 & n is prime holds
not n divides 2011 by XPRIMET1:28;
hence 2011 is prime by NAT_4:14; :: thesis: