5471 = (2 * 2735) + 1 ;
hence not 2 divides 5471 by NAT_4:9; :: thesis:
5471 = (3 * 1823) + 2 ;
hence not 3 divides 5471 by NAT_4:9; :: thesis:
5471 = (5 * 1094) + 1 ;
hence not 5 divides 5471 by NAT_4:9; :: thesis:
5471 = (7 * 781) + 4 ;
hence not 7 divides 5471 by NAT_4:9; :: thesis:
5471 = (11 * 497) + 4 ;
hence not 11 divides 5471 by NAT_4:9; :: thesis:
5471 = (13 * 420) + 11 ;
hence not 13 divides 5471 by NAT_4:9; :: thesis:
5471 = (17 * 321) + 14 ;
hence not 17 divides 5471 by NAT_4:9; :: thesis:
5471 = (19 * 287) + 18 ;
hence not 19 divides 5471 by NAT_4:9; :: thesis:
5471 = (23 * 237) + 20 ;
hence not 23 divides 5471 by NAT_4:9; :: thesis:
5471 = (29 * 188) + 19 ;
hence not 29 divides 5471 by NAT_4:9; :: thesis:
5471 = (31 * 176) + 15 ;
hence not 31 divides 5471 by NAT_4:9; :: thesis:
5471 = (37 * 147) + 32 ;
hence not 37 divides 5471 by NAT_4:9; :: thesis:
5471 = (41 * 133) + 18 ;
hence not 41 divides 5471 by NAT_4:9; :: thesis:
5471 = (43 * 127) + 10 ;
hence not 43 divides 5471 by NAT_4:9; :: thesis:
5471 = (47 * 116) + 19 ;
hence not 47 divides 5471 by NAT_4:9; :: thesis:
5471 = (53 * 103) + 12 ;
hence not 53 divides 5471 by NAT_4:9; :: thesis:
5471 = (59 * 92) + 43 ;
hence not 59 divides 5471 by NAT_4:9; :: thesis:
5471 = (61 * 89) + 42 ;
hence not 61 divides 5471 by NAT_4:9; :: thesis:
5471 = (67 * 81) + 44 ;
hence not 67 divides 5471 by NAT_4:9; :: thesis:
5471 = (71 * 77) + 4 ;
hence not 71 divides 5471 by NAT_4:9; :: thesis:
5471 = (73 * 74) + 69 ;
hence not 73 divides 5471 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5471 & n is prime holds
not n divides 5471 by XPRIMET1:42;
hence 5471 is prime by NAT_4:14; :: thesis: