691 = (2 * 345) + 1 ;
hence not 2 divides 691 by NAT_4:9; :: thesis:
691 = (3 * 230) + 1 ;
hence not 3 divides 691 by NAT_4:9; :: thesis:
691 = (5 * 138) + 1 ;
hence not 5 divides 691 by NAT_4:9; :: thesis:
691 = (7 * 98) + 5 ;
hence not 7 divides 691 by NAT_4:9; :: thesis:
691 = (11 * 62) + 9 ;
hence not 11 divides 691 by NAT_4:9; :: thesis:
691 = (13 * 53) + 2 ;
hence not 13 divides 691 by NAT_4:9; :: thesis:
691 = (17 * 40) + 11 ;
hence not 17 divides 691 by NAT_4:9; :: thesis:
691 = (19 * 36) + 7 ;
hence not 19 divides 691 by NAT_4:9; :: thesis:
691 = (23 * 30) + 1 ;
hence not 23 divides 691 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 691 & n is prime holds
not n divides 691 by XPRIMET1:18;
hence 691 is prime by NAT_4:14; :: thesis: