1931 = (2 * 965) + 1 ;
hence not 2 divides 1931 by NAT_4:9; :: thesis:
1931 = (3 * 643) + 2 ;
hence not 3 divides 1931 by NAT_4:9; :: thesis:
1931 = (5 * 386) + 1 ;
hence not 5 divides 1931 by NAT_4:9; :: thesis:
1931 = (7 * 275) + 6 ;
hence not 7 divides 1931 by NAT_4:9; :: thesis:
1931 = (11 * 175) + 6 ;
hence not 11 divides 1931 by NAT_4:9; :: thesis:
1931 = (13 * 148) + 7 ;
hence not 13 divides 1931 by NAT_4:9; :: thesis:
1931 = (17 * 113) + 10 ;
hence not 17 divides 1931 by NAT_4:9; :: thesis:
1931 = (19 * 101) + 12 ;
hence not 19 divides 1931 by NAT_4:9; :: thesis:
1931 = (23 * 83) + 22 ;
hence not 23 divides 1931 by NAT_4:9; :: thesis:
1931 = (29 * 66) + 17 ;
hence not 29 divides 1931 by NAT_4:9; :: thesis:
1931 = (31 * 62) + 9 ;
hence not 31 divides 1931 by NAT_4:9; :: thesis:
1931 = (37 * 52) + 7 ;
hence not 37 divides 1931 by NAT_4:9; :: thesis:
1931 = (41 * 47) + 4 ;
hence not 41 divides 1931 by NAT_4:9; :: thesis:
1931 = (43 * 44) + 39 ;
hence not 43 divides 1931 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1931 & n is prime holds
not n divides 1931 by XPRIMET1:28;
hence 1931 is prime by NAT_4:14; :: thesis: