5119 = (2 * 2559) + 1 ;
hence not 2 divides 5119 by NAT_4:9; :: thesis:
5119 = (3 * 1706) + 1 ;
hence not 3 divides 5119 by NAT_4:9; :: thesis:
5119 = (5 * 1023) + 4 ;
hence not 5 divides 5119 by NAT_4:9; :: thesis:
5119 = (7 * 731) + 2 ;
hence not 7 divides 5119 by NAT_4:9; :: thesis:
5119 = (11 * 465) + 4 ;
hence not 11 divides 5119 by NAT_4:9; :: thesis:
5119 = (13 * 393) + 10 ;
hence not 13 divides 5119 by NAT_4:9; :: thesis:
5119 = (17 * 301) + 2 ;
hence not 17 divides 5119 by NAT_4:9; :: thesis:
5119 = (19 * 269) + 8 ;
hence not 19 divides 5119 by NAT_4:9; :: thesis:
5119 = (23 * 222) + 13 ;
hence not 23 divides 5119 by NAT_4:9; :: thesis:
5119 = (29 * 176) + 15 ;
hence not 29 divides 5119 by NAT_4:9; :: thesis:
5119 = (31 * 165) + 4 ;
hence not 31 divides 5119 by NAT_4:9; :: thesis:
5119 = (37 * 138) + 13 ;
hence not 37 divides 5119 by NAT_4:9; :: thesis:
5119 = (41 * 124) + 35 ;
hence not 41 divides 5119 by NAT_4:9; :: thesis:
5119 = (43 * 119) + 2 ;
hence not 43 divides 5119 by NAT_4:9; :: thesis:
5119 = (47 * 108) + 43 ;
hence not 47 divides 5119 by NAT_4:9; :: thesis:
5119 = (53 * 96) + 31 ;
hence not 53 divides 5119 by NAT_4:9; :: thesis:
5119 = (59 * 86) + 45 ;
hence not 59 divides 5119 by NAT_4:9; :: thesis:
5119 = (61 * 83) + 56 ;
hence not 61 divides 5119 by NAT_4:9; :: thesis:
5119 = (67 * 76) + 27 ;
hence not 67 divides 5119 by NAT_4:9; :: thesis:
5119 = (71 * 72) + 7 ;
hence not 71 divides 5119 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5119 & n is prime holds
not n divides 5119 by XPRIMET1:40;
hence 5119 is prime by NAT_4:14; :: thesis: