5779 = (2 * 2889) + 1 ;
hence not 2 divides 5779 by NAT_4:9; :: thesis:
5779 = (3 * 1926) + 1 ;
hence not 3 divides 5779 by NAT_4:9; :: thesis:
5779 = (5 * 1155) + 4 ;
hence not 5 divides 5779 by NAT_4:9; :: thesis:
5779 = (7 * 825) + 4 ;
hence not 7 divides 5779 by NAT_4:9; :: thesis:
5779 = (11 * 525) + 4 ;
hence not 11 divides 5779 by NAT_4:9; :: thesis:
5779 = (13 * 444) + 7 ;
hence not 13 divides 5779 by NAT_4:9; :: thesis:
5779 = (17 * 339) + 16 ;
hence not 17 divides 5779 by NAT_4:9; :: thesis:
5779 = (19 * 304) + 3 ;
hence not 19 divides 5779 by NAT_4:9; :: thesis:
5779 = (23 * 251) + 6 ;
hence not 23 divides 5779 by NAT_4:9; :: thesis:
5779 = (29 * 199) + 8 ;
hence not 29 divides 5779 by NAT_4:9; :: thesis:
5779 = (31 * 186) + 13 ;
hence not 31 divides 5779 by NAT_4:9; :: thesis:
5779 = (37 * 156) + 7 ;
hence not 37 divides 5779 by NAT_4:9; :: thesis:
5779 = (41 * 140) + 39 ;
hence not 41 divides 5779 by NAT_4:9; :: thesis:
5779 = (43 * 134) + 17 ;
hence not 43 divides 5779 by NAT_4:9; :: thesis:
5779 = (47 * 122) + 45 ;
hence not 47 divides 5779 by NAT_4:9; :: thesis:
5779 = (53 * 109) + 2 ;
hence not 53 divides 5779 by NAT_4:9; :: thesis:
5779 = (59 * 97) + 56 ;
hence not 59 divides 5779 by NAT_4:9; :: thesis:
5779 = (61 * 94) + 45 ;
hence not 61 divides 5779 by NAT_4:9; :: thesis:
5779 = (67 * 86) + 17 ;
hence not 67 divides 5779 by NAT_4:9; :: thesis:
5779 = (71 * 81) + 28 ;
hence not 71 divides 5779 by NAT_4:9; :: thesis:
5779 = (73 * 79) + 12 ;
hence not 73 divides 5779 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5779 & n is prime holds
not n divides 5779 by XPRIMET1:42;
hence 5779 is prime by NAT_4:14; :: thesis: