1871 = (2 * 935) + 1 ;
hence not 2 divides 1871 by NAT_4:9; :: thesis:
1871 = (3 * 623) + 2 ;
hence not 3 divides 1871 by NAT_4:9; :: thesis:
1871 = (5 * 374) + 1 ;
hence not 5 divides 1871 by NAT_4:9; :: thesis:
1871 = (7 * 267) + 2 ;
hence not 7 divides 1871 by NAT_4:9; :: thesis:
1871 = (11 * 170) + 1 ;
hence not 11 divides 1871 by NAT_4:9; :: thesis:
1871 = (13 * 143) + 12 ;
hence not 13 divides 1871 by NAT_4:9; :: thesis:
1871 = (17 * 110) + 1 ;
hence not 17 divides 1871 by NAT_4:9; :: thesis:
1871 = (19 * 98) + 9 ;
hence not 19 divides 1871 by NAT_4:9; :: thesis:
1871 = (23 * 81) + 8 ;
hence not 23 divides 1871 by NAT_4:9; :: thesis:
1871 = (29 * 64) + 15 ;
hence not 29 divides 1871 by NAT_4:9; :: thesis:
1871 = (31 * 60) + 11 ;
hence not 31 divides 1871 by NAT_4:9; :: thesis:
1871 = (37 * 50) + 21 ;
hence not 37 divides 1871 by NAT_4:9; :: thesis:
1871 = (41 * 45) + 26 ;
hence not 41 divides 1871 by NAT_4:9; :: thesis:
1871 = (43 * 43) + 22 ;
hence not 43 divides 1871 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1871 & n is prime holds
not n divides 1871 by XPRIMET1:28;
hence 1871 is prime by NAT_4:14; :: thesis: