1583 = (2 * 791) + 1 ;
hence not 2 divides 1583 by NAT_4:9; :: thesis:
1583 = (3 * 527) + 2 ;
hence not 3 divides 1583 by NAT_4:9; :: thesis:
1583 = (5 * 316) + 3 ;
hence not 5 divides 1583 by NAT_4:9; :: thesis:
1583 = (7 * 226) + 1 ;
hence not 7 divides 1583 by NAT_4:9; :: thesis:
1583 = (11 * 143) + 10 ;
hence not 11 divides 1583 by NAT_4:9; :: thesis:
1583 = (13 * 121) + 10 ;
hence not 13 divides 1583 by NAT_4:9; :: thesis:
1583 = (17 * 93) + 2 ;
hence not 17 divides 1583 by NAT_4:9; :: thesis:
1583 = (19 * 83) + 6 ;
hence not 19 divides 1583 by NAT_4:9; :: thesis:
1583 = (23 * 68) + 19 ;
hence not 23 divides 1583 by NAT_4:9; :: thesis:
1583 = (29 * 54) + 17 ;
hence not 29 divides 1583 by NAT_4:9; :: thesis:
1583 = (31 * 51) + 2 ;
hence not 31 divides 1583 by NAT_4:9; :: thesis:
1583 = (37 * 42) + 29 ;
hence not 37 divides 1583 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1583 & n is prime holds
not n divides 1583 by XPRIMET1:24;
hence 1583 is prime by NAT_4:14; :: thesis: