1759 = (2 * 879) + 1 ;
hence not 2 divides 1759 by NAT_4:9; :: thesis:
1759 = (3 * 586) + 1 ;
hence not 3 divides 1759 by NAT_4:9; :: thesis:
1759 = (5 * 351) + 4 ;
hence not 5 divides 1759 by NAT_4:9; :: thesis:
1759 = (7 * 251) + 2 ;
hence not 7 divides 1759 by NAT_4:9; :: thesis:
1759 = (11 * 159) + 10 ;
hence not 11 divides 1759 by NAT_4:9; :: thesis:
1759 = (13 * 135) + 4 ;
hence not 13 divides 1759 by NAT_4:9; :: thesis:
1759 = (17 * 103) + 8 ;
hence not 17 divides 1759 by NAT_4:9; :: thesis:
1759 = (19 * 92) + 11 ;
hence not 19 divides 1759 by NAT_4:9; :: thesis:
1759 = (23 * 76) + 11 ;
hence not 23 divides 1759 by NAT_4:9; :: thesis:
1759 = (29 * 60) + 19 ;
hence not 29 divides 1759 by NAT_4:9; :: thesis:
1759 = (31 * 56) + 23 ;
hence not 31 divides 1759 by NAT_4:9; :: thesis:
1759 = (37 * 47) + 20 ;
hence not 37 divides 1759 by NAT_4:9; :: thesis:
1759 = (41 * 42) + 37 ;
hence not 41 divides 1759 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1759 & n is prime holds
not n divides 1759 by XPRIMET1:26;
hence 1759 is prime by NAT_4:14; :: thesis: