1861 = (2 * 930) + 1 ;
hence not 2 divides 1861 by NAT_4:9; :: thesis:
1861 = (3 * 620) + 1 ;
hence not 3 divides 1861 by NAT_4:9; :: thesis:
1861 = (5 * 372) + 1 ;
hence not 5 divides 1861 by NAT_4:9; :: thesis:
1861 = (7 * 265) + 6 ;
hence not 7 divides 1861 by NAT_4:9; :: thesis:
1861 = (11 * 169) + 2 ;
hence not 11 divides 1861 by NAT_4:9; :: thesis:
1861 = (13 * 143) + 2 ;
hence not 13 divides 1861 by NAT_4:9; :: thesis:
1861 = (17 * 109) + 8 ;
hence not 17 divides 1861 by NAT_4:9; :: thesis:
1861 = (19 * 97) + 18 ;
hence not 19 divides 1861 by NAT_4:9; :: thesis:
1861 = (23 * 80) + 21 ;
hence not 23 divides 1861 by NAT_4:9; :: thesis:
1861 = (29 * 64) + 5 ;
hence not 29 divides 1861 by NAT_4:9; :: thesis:
1861 = (31 * 60) + 1 ;
hence not 31 divides 1861 by NAT_4:9; :: thesis:
1861 = (37 * 50) + 11 ;
hence not 37 divides 1861 by NAT_4:9; :: thesis:
1861 = (41 * 45) + 16 ;
hence not 41 divides 1861 by NAT_4:9; :: thesis:
1861 = (43 * 43) + 12 ;
hence not 43 divides 1861 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1861 & n is prime holds
not n divides 1861 by XPRIMET1:28;
hence 1861 is prime by NAT_4:14; :: thesis: