now :: thesis: ( not 2 divides 2657 & not 3 divides 2657 & not 5 divides 2657 & not 7 divides 2657 & not 11 divides 2657 & not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )

2657 = (2 * 1328) + 1 ;
hence not 2 divides 2657 by NAT_4:9; :: thesis: ( not 3 divides 2657 & not 5 divides 2657 & not 7 divides 2657 & not 11 divides 2657 & not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (3 * 885) + 2 ;
hence not 3 divides 2657 by NAT_4:9; :: thesis: ( not 5 divides 2657 & not 7 divides 2657 & not 11 divides 2657 & not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (5 * 531) + 2 ;
hence not 5 divides 2657 by NAT_4:9; :: thesis: ( not 7 divides 2657 & not 11 divides 2657 & not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (7 * 379) + 4 ;
hence not 7 divides 2657 by NAT_4:9; :: thesis: ( not 11 divides 2657 & not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (11 * 241) + 6 ;
hence not 11 divides 2657 by NAT_4:9; :: thesis: ( not 13 divides 2657 & not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (13 * 204) + 5 ;
hence not 13 divides 2657 by NAT_4:9; :: thesis: ( not 17 divides 2657 & not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (17 * 156) + 5 ;
hence not 17 divides 2657 by NAT_4:9; :: thesis: ( not 19 divides 2657 & not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (19 * 139) + 16 ;
hence not 19 divides 2657 by NAT_4:9; :: thesis: ( not 23 divides 2657 & not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (23 * 115) + 12 ;
hence not 23 divides 2657 by NAT_4:9; :: thesis: ( not 29 divides 2657 & not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (29 * 91) + 18 ;
hence not 29 divides 2657 by NAT_4:9; :: thesis: ( not 31 divides 2657 & not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (31 * 85) + 22 ;
hence not 31 divides 2657 by NAT_4:9; :: thesis: ( not 37 divides 2657 & not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (37 * 71) + 30 ;
hence not 37 divides 2657 by NAT_4:9; :: thesis: ( not 41 divides 2657 & not 43 divides 2657 & not 47 divides 2657 )
2657 = (41 * 64) + 33 ;
hence not 41 divides 2657 by NAT_4:9; :: thesis: ( not 43 divides 2657 & not 47 divides 2657 )
2657 = (43 * 61) + 34 ;
hence not 43 divides 2657 by NAT_4:9; :: thesis: not 47 divides 2657
2657 = (47 * 56) + 25 ;
hence not 47 divides 2657 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2657 & n is prime holds
not n divides 2657 by XPRIMET1:30;
hence 2657 is prime by NAT_4:14; :: thesis: verum