1801 = (2 * 900) + 1 ;
hence not 2 divides 1801 by NAT_4:9; :: thesis:
1801 = (3 * 600) + 1 ;
hence not 3 divides 1801 by NAT_4:9; :: thesis:
1801 = (5 * 360) + 1 ;
hence not 5 divides 1801 by NAT_4:9; :: thesis:
1801 = (7 * 257) + 2 ;
hence not 7 divides 1801 by NAT_4:9; :: thesis:
1801 = (11 * 163) + 8 ;
hence not 11 divides 1801 by NAT_4:9; :: thesis:
1801 = (13 * 138) + 7 ;
hence not 13 divides 1801 by NAT_4:9; :: thesis:
1801 = (17 * 105) + 16 ;
hence not 17 divides 1801 by NAT_4:9; :: thesis:
1801 = (19 * 94) + 15 ;
hence not 19 divides 1801 by NAT_4:9; :: thesis:
1801 = (23 * 78) + 7 ;
hence not 23 divides 1801 by NAT_4:9; :: thesis:
1801 = (29 * 62) + 3 ;
hence not 29 divides 1801 by NAT_4:9; :: thesis:
1801 = (31 * 58) + 3 ;
hence not 31 divides 1801 by NAT_4:9; :: thesis:
1801 = (37 * 48) + 25 ;
hence not 37 divides 1801 by NAT_4:9; :: thesis:
1801 = (41 * 43) + 38 ;
hence not 41 divides 1801 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1801 & n is prime holds
not n divides 1801 by XPRIMET1:26;
hence 1801 is prime by NAT_4:14; :: thesis: