1579 = (2 * 789) + 1 ;
hence not 2 divides 1579 by NAT_4:9; :: thesis:
1579 = (3 * 526) + 1 ;
hence not 3 divides 1579 by NAT_4:9; :: thesis:
1579 = (5 * 315) + 4 ;
hence not 5 divides 1579 by NAT_4:9; :: thesis:
1579 = (7 * 225) + 4 ;
hence not 7 divides 1579 by NAT_4:9; :: thesis:
1579 = (11 * 143) + 6 ;
hence not 11 divides 1579 by NAT_4:9; :: thesis:
1579 = (13 * 121) + 6 ;
hence not 13 divides 1579 by NAT_4:9; :: thesis:
1579 = (17 * 92) + 15 ;
hence not 17 divides 1579 by NAT_4:9; :: thesis:
1579 = (19 * 83) + 2 ;
hence not 19 divides 1579 by NAT_4:9; :: thesis:
1579 = (23 * 68) + 15 ;
hence not 23 divides 1579 by NAT_4:9; :: thesis:
1579 = (29 * 54) + 13 ;
hence not 29 divides 1579 by NAT_4:9; :: thesis:
1579 = (31 * 50) + 29 ;
hence not 31 divides 1579 by NAT_4:9; :: thesis:
1579 = (37 * 42) + 25 ;
hence not 37 divides 1579 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1579 & n is prime holds
not n divides 1579 by XPRIMET1:24;
hence 1579 is prime by NAT_4:14; :: thesis: