now :: thesis: ( not 2 divides 1997 & not 3 divides 1997 & not 5 divides 1997 & not 7 divides 1997 & not 11 divides 1997 & not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )

1997 = (2 * 998) + 1 ;
hence not 2 divides 1997 by NAT_4:9; :: thesis: ( not 3 divides 1997 & not 5 divides 1997 & not 7 divides 1997 & not 11 divides 1997 & not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (3 * 665) + 2 ;
hence not 3 divides 1997 by NAT_4:9; :: thesis: ( not 5 divides 1997 & not 7 divides 1997 & not 11 divides 1997 & not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (5 * 399) + 2 ;
hence not 5 divides 1997 by NAT_4:9; :: thesis: ( not 7 divides 1997 & not 11 divides 1997 & not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (7 * 285) + 2 ;
hence not 7 divides 1997 by NAT_4:9; :: thesis: ( not 11 divides 1997 & not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (11 * 181) + 6 ;
hence not 11 divides 1997 by NAT_4:9; :: thesis: ( not 13 divides 1997 & not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (13 * 153) + 8 ;
hence not 13 divides 1997 by NAT_4:9; :: thesis: ( not 17 divides 1997 & not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (17 * 117) + 8 ;
hence not 17 divides 1997 by NAT_4:9; :: thesis: ( not 19 divides 1997 & not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (19 * 105) + 2 ;
hence not 19 divides 1997 by NAT_4:9; :: thesis: ( not 23 divides 1997 & not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (23 * 86) + 19 ;
hence not 23 divides 1997 by NAT_4:9; :: thesis: ( not 29 divides 1997 & not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (29 * 68) + 25 ;
hence not 29 divides 1997 by NAT_4:9; :: thesis: ( not 31 divides 1997 & not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (31 * 64) + 13 ;
hence not 31 divides 1997 by NAT_4:9; :: thesis: ( not 37 divides 1997 & not 41 divides 1997 & not 43 divides 1997 )
1997 = (37 * 53) + 36 ;
hence not 37 divides 1997 by NAT_4:9; :: thesis: ( not 41 divides 1997 & not 43 divides 1997 )
1997 = (41 * 48) + 29 ;
hence not 41 divides 1997 by NAT_4:9; :: thesis: not 43 divides 1997
1997 = (43 * 46) + 19 ;
hence not 43 divides 1997 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1997 & n is prime holds
not n divides 1997 by XPRIMET1:28;
hence 1997 is prime by NAT_4:14; :: thesis: verum