5431 = (2 * 2715) + 1 ;
hence not 2 divides 5431 by NAT_4:9; :: thesis:
5431 = (3 * 1810) + 1 ;
hence not 3 divides 5431 by NAT_4:9; :: thesis:
5431 = (5 * 1086) + 1 ;
hence not 5 divides 5431 by NAT_4:9; :: thesis:
5431 = (7 * 775) + 6 ;
hence not 7 divides 5431 by NAT_4:9; :: thesis:
5431 = (11 * 493) + 8 ;
hence not 11 divides 5431 by NAT_4:9; :: thesis:
5431 = (13 * 417) + 10 ;
hence not 13 divides 5431 by NAT_4:9; :: thesis:
5431 = (17 * 319) + 8 ;
hence not 17 divides 5431 by NAT_4:9; :: thesis:
5431 = (19 * 285) + 16 ;
hence not 19 divides 5431 by NAT_4:9; :: thesis:
5431 = (23 * 236) + 3 ;
hence not 23 divides 5431 by NAT_4:9; :: thesis:
5431 = (29 * 187) + 8 ;
hence not 29 divides 5431 by NAT_4:9; :: thesis:
5431 = (31 * 175) + 6 ;
hence not 31 divides 5431 by NAT_4:9; :: thesis:
5431 = (37 * 146) + 29 ;
hence not 37 divides 5431 by NAT_4:9; :: thesis:
5431 = (41 * 132) + 19 ;
hence not 41 divides 5431 by NAT_4:9; :: thesis:
5431 = (43 * 126) + 13 ;
hence not 43 divides 5431 by NAT_4:9; :: thesis:
5431 = (47 * 115) + 26 ;
hence not 47 divides 5431 by NAT_4:9; :: thesis:
5431 = (53 * 102) + 25 ;
hence not 53 divides 5431 by NAT_4:9; :: thesis:
5431 = (59 * 92) + 3 ;
hence not 59 divides 5431 by NAT_4:9; :: thesis:
5431 = (61 * 89) + 2 ;
hence not 61 divides 5431 by NAT_4:9; :: thesis:
5431 = (67 * 81) + 4 ;
hence not 67 divides 5431 by NAT_4:9; :: thesis:
5431 = (71 * 76) + 35 ;
hence not 71 divides 5431 by NAT_4:9; :: thesis:
5431 = (73 * 74) + 29 ;
hence not 73 divides 5431 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5431 & n is prime holds
not n divides 5431 by XPRIMET1:42;
hence 5431 is prime by NAT_4:14; :: thesis: