5399 = (2 * 2699) + 1 ;
hence not 2 divides 5399 by NAT_4:9; :: thesis:
5399 = (3 * 1799) + 2 ;
hence not 3 divides 5399 by NAT_4:9; :: thesis:
5399 = (5 * 1079) + 4 ;
hence not 5 divides 5399 by NAT_4:9; :: thesis:
5399 = (7 * 771) + 2 ;
hence not 7 divides 5399 by NAT_4:9; :: thesis:
5399 = (11 * 490) + 9 ;
hence not 11 divides 5399 by NAT_4:9; :: thesis:
5399 = (13 * 415) + 4 ;
hence not 13 divides 5399 by NAT_4:9; :: thesis:
5399 = (17 * 317) + 10 ;
hence not 17 divides 5399 by NAT_4:9; :: thesis:
5399 = (19 * 284) + 3 ;
hence not 19 divides 5399 by NAT_4:9; :: thesis:
5399 = (23 * 234) + 17 ;
hence not 23 divides 5399 by NAT_4:9; :: thesis:
5399 = (29 * 186) + 5 ;
hence not 29 divides 5399 by NAT_4:9; :: thesis:
5399 = (31 * 174) + 5 ;
hence not 31 divides 5399 by NAT_4:9; :: thesis:
5399 = (37 * 145) + 34 ;
hence not 37 divides 5399 by NAT_4:9; :: thesis:
5399 = (41 * 131) + 28 ;
hence not 41 divides 5399 by NAT_4:9; :: thesis:
5399 = (43 * 125) + 24 ;
hence not 43 divides 5399 by NAT_4:9; :: thesis:
5399 = (47 * 114) + 41 ;
hence not 47 divides 5399 by NAT_4:9; :: thesis:
5399 = (53 * 101) + 46 ;
hence not 53 divides 5399 by NAT_4:9; :: thesis:
5399 = (59 * 91) + 30 ;
hence not 59 divides 5399 by NAT_4:9; :: thesis:
5399 = (61 * 88) + 31 ;
hence not 61 divides 5399 by NAT_4:9; :: thesis:
5399 = (67 * 80) + 39 ;
hence not 67 divides 5399 by NAT_4:9; :: thesis:
5399 = (71 * 76) + 3 ;
hence not 71 divides 5399 by NAT_4:9; :: thesis:
5399 = (73 * 73) + 70 ;
hence not 73 divides 5399 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5399 & n is prime holds
not n divides 5399 by XPRIMET1:42;
hence 5399 is prime by NAT_4:14; :: thesis: