1693 = (2 * 846) + 1 ;
hence not 2 divides 1693 by NAT_4:9; :: thesis:
1693 = (3 * 564) + 1 ;
hence not 3 divides 1693 by NAT_4:9; :: thesis:
1693 = (5 * 338) + 3 ;
hence not 5 divides 1693 by NAT_4:9; :: thesis:
1693 = (7 * 241) + 6 ;
hence not 7 divides 1693 by NAT_4:9; :: thesis:
1693 = (11 * 153) + 10 ;
hence not 11 divides 1693 by NAT_4:9; :: thesis:
1693 = (13 * 130) + 3 ;
hence not 13 divides 1693 by NAT_4:9; :: thesis:
1693 = (17 * 99) + 10 ;
hence not 17 divides 1693 by NAT_4:9; :: thesis:
1693 = (19 * 89) + 2 ;
hence not 19 divides 1693 by NAT_4:9; :: thesis:
1693 = (23 * 73) + 14 ;
hence not 23 divides 1693 by NAT_4:9; :: thesis:
1693 = (29 * 58) + 11 ;
hence not 29 divides 1693 by NAT_4:9; :: thesis:
1693 = (31 * 54) + 19 ;
hence not 31 divides 1693 by NAT_4:9; :: thesis:
1693 = (37 * 45) + 28 ;
hence not 37 divides 1693 by NAT_4:9; :: thesis:
1693 = (41 * 41) + 12 ;
hence not 41 divides 1693 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1693 & n is prime holds
not n divides 1693 by XPRIMET1:26;
hence 1693 is prime by NAT_4:14; :: thesis: