now :: thesis: ( not 2 divides 1811 & not 3 divides 1811 & not 5 divides 1811 & not 7 divides 1811 & not 11 divides 1811 & not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )

1811 = (2 * 905) + 1 ;
hence not 2 divides 1811 by NAT_4:9; :: thesis: ( not 3 divides 1811 & not 5 divides 1811 & not 7 divides 1811 & not 11 divides 1811 & not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (3 * 603) + 2 ;
hence not 3 divides 1811 by NAT_4:9; :: thesis: ( not 5 divides 1811 & not 7 divides 1811 & not 11 divides 1811 & not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (5 * 362) + 1 ;
hence not 5 divides 1811 by NAT_4:9; :: thesis: ( not 7 divides 1811 & not 11 divides 1811 & not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (7 * 258) + 5 ;
hence not 7 divides 1811 by NAT_4:9; :: thesis: ( not 11 divides 1811 & not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (11 * 164) + 7 ;
hence not 11 divides 1811 by NAT_4:9; :: thesis: ( not 13 divides 1811 & not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (13 * 139) + 4 ;
hence not 13 divides 1811 by NAT_4:9; :: thesis: ( not 17 divides 1811 & not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (17 * 106) + 9 ;
hence not 17 divides 1811 by NAT_4:9; :: thesis: ( not 19 divides 1811 & not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (19 * 95) + 6 ;
hence not 19 divides 1811 by NAT_4:9; :: thesis: ( not 23 divides 1811 & not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (23 * 78) + 17 ;
hence not 23 divides 1811 by NAT_4:9; :: thesis: ( not 29 divides 1811 & not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (29 * 62) + 13 ;
hence not 29 divides 1811 by NAT_4:9; :: thesis: ( not 31 divides 1811 & not 37 divides 1811 & not 41 divides 1811 )
1811 = (31 * 58) + 13 ;
hence not 31 divides 1811 by NAT_4:9; :: thesis: ( not 37 divides 1811 & not 41 divides 1811 )
1811 = (37 * 48) + 35 ;
hence not 37 divides 1811 by NAT_4:9; :: thesis: not 41 divides 1811
1811 = (41 * 44) + 7 ;
hence not 41 divides 1811 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1811 & n is prime holds
not n divides 1811 by XPRIMET1:26;
hence 1811 is prime by NAT_4:14; :: thesis: verum