now :: thesis: ( not 2 divides 983 & not 3 divides 983 & not 5 divides 983 & not 7 divides 983 & not 11 divides 983 & not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )

983 = (2 * 491) + 1 ;
hence not 2 divides 983 by NAT_4:9; :: thesis: ( not 3 divides 983 & not 5 divides 983 & not 7 divides 983 & not 11 divides 983 & not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (3 * 327) + 2 ;
hence not 3 divides 983 by NAT_4:9; :: thesis: ( not 5 divides 983 & not 7 divides 983 & not 11 divides 983 & not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (5 * 196) + 3 ;
hence not 5 divides 983 by NAT_4:9; :: thesis: ( not 7 divides 983 & not 11 divides 983 & not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (7 * 140) + 3 ;
hence not 7 divides 983 by NAT_4:9; :: thesis: ( not 11 divides 983 & not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (11 * 89) + 4 ;
hence not 11 divides 983 by NAT_4:9; :: thesis: ( not 13 divides 983 & not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (13 * 75) + 8 ;
hence not 13 divides 983 by NAT_4:9; :: thesis: ( not 17 divides 983 & not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (17 * 57) + 14 ;
hence not 17 divides 983 by NAT_4:9; :: thesis: ( not 19 divides 983 & not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (19 * 51) + 14 ;
hence not 19 divides 983 by NAT_4:9; :: thesis: ( not 23 divides 983 & not 29 divides 983 & not 31 divides 983 )
983 = (23 * 42) + 17 ;
hence not 23 divides 983 by NAT_4:9; :: thesis: ( not 29 divides 983 & not 31 divides 983 )
983 = (29 * 33) + 26 ;
hence not 29 divides 983 by NAT_4:9; :: thesis: not 31 divides 983
983 = (31 * 31) + 22 ;
hence not 31 divides 983 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 983 & n is prime holds
not n divides 983 by XPRIMET1:22;
hence 983 is prime by NAT_4:14; :: thesis: verum