now :: thesis: ( not 2 divides 2671 & not 3 divides 2671 & not 5 divides 2671 & not 7 divides 2671 & not 11 divides 2671 & not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )

2671 = (2 * 1335) + 1 ;
hence not 2 divides 2671 by NAT_4:9; :: thesis: ( not 3 divides 2671 & not 5 divides 2671 & not 7 divides 2671 & not 11 divides 2671 & not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (3 * 890) + 1 ;
hence not 3 divides 2671 by NAT_4:9; :: thesis: ( not 5 divides 2671 & not 7 divides 2671 & not 11 divides 2671 & not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (5 * 534) + 1 ;
hence not 5 divides 2671 by NAT_4:9; :: thesis: ( not 7 divides 2671 & not 11 divides 2671 & not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (7 * 381) + 4 ;
hence not 7 divides 2671 by NAT_4:9; :: thesis: ( not 11 divides 2671 & not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (11 * 242) + 9 ;
hence not 11 divides 2671 by NAT_4:9; :: thesis: ( not 13 divides 2671 & not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (13 * 205) + 6 ;
hence not 13 divides 2671 by NAT_4:9; :: thesis: ( not 17 divides 2671 & not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (17 * 157) + 2 ;
hence not 17 divides 2671 by NAT_4:9; :: thesis: ( not 19 divides 2671 & not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (19 * 140) + 11 ;
hence not 19 divides 2671 by NAT_4:9; :: thesis: ( not 23 divides 2671 & not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (23 * 116) + 3 ;
hence not 23 divides 2671 by NAT_4:9; :: thesis: ( not 29 divides 2671 & not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (29 * 92) + 3 ;
hence not 29 divides 2671 by NAT_4:9; :: thesis: ( not 31 divides 2671 & not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (31 * 86) + 5 ;
hence not 31 divides 2671 by NAT_4:9; :: thesis: ( not 37 divides 2671 & not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (37 * 72) + 7 ;
hence not 37 divides 2671 by NAT_4:9; :: thesis: ( not 41 divides 2671 & not 43 divides 2671 & not 47 divides 2671 )
2671 = (41 * 65) + 6 ;
hence not 41 divides 2671 by NAT_4:9; :: thesis: ( not 43 divides 2671 & not 47 divides 2671 )
2671 = (43 * 62) + 5 ;
hence not 43 divides 2671 by NAT_4:9; :: thesis: not 47 divides 2671
2671 = (47 * 56) + 39 ;
hence not 47 divides 2671 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2671 & n is prime holds
not n divides 2671 by XPRIMET1:30;
hence 2671 is prime by NAT_4:14; :: thesis: verum