5711 = (2 * 2855) + 1 ;
hence not 2 divides 5711 by NAT_4:9; :: thesis:
5711 = (3 * 1903) + 2 ;
hence not 3 divides 5711 by NAT_4:9; :: thesis:
5711 = (5 * 1142) + 1 ;
hence not 5 divides 5711 by NAT_4:9; :: thesis:
5711 = (7 * 815) + 6 ;
hence not 7 divides 5711 by NAT_4:9; :: thesis:
5711 = (11 * 519) + 2 ;
hence not 11 divides 5711 by NAT_4:9; :: thesis:
5711 = (13 * 439) + 4 ;
hence not 13 divides 5711 by NAT_4:9; :: thesis:
5711 = (17 * 335) + 16 ;
hence not 17 divides 5711 by NAT_4:9; :: thesis:
5711 = (19 * 300) + 11 ;
hence not 19 divides 5711 by NAT_4:9; :: thesis:
5711 = (23 * 248) + 7 ;
hence not 23 divides 5711 by NAT_4:9; :: thesis:
5711 = (29 * 196) + 27 ;
hence not 29 divides 5711 by NAT_4:9; :: thesis:
5711 = (31 * 184) + 7 ;
hence not 31 divides 5711 by NAT_4:9; :: thesis:
5711 = (37 * 154) + 13 ;
hence not 37 divides 5711 by NAT_4:9; :: thesis:
5711 = (41 * 139) + 12 ;
hence not 41 divides 5711 by NAT_4:9; :: thesis:
5711 = (43 * 132) + 35 ;
hence not 43 divides 5711 by NAT_4:9; :: thesis:
5711 = (47 * 121) + 24 ;
hence not 47 divides 5711 by NAT_4:9; :: thesis:
5711 = (53 * 107) + 40 ;
hence not 53 divides 5711 by NAT_4:9; :: thesis:
5711 = (59 * 96) + 47 ;
hence not 59 divides 5711 by NAT_4:9; :: thesis:
5711 = (61 * 93) + 38 ;
hence not 61 divides 5711 by NAT_4:9; :: thesis:
5711 = (67 * 85) + 16 ;
hence not 67 divides 5711 by NAT_4:9; :: thesis:
5711 = (71 * 80) + 31 ;
hence not 71 divides 5711 by NAT_4:9; :: thesis:
5711 = (73 * 78) + 17 ;
hence not 73 divides 5711 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5711 & n is prime holds
not n divides 5711 by XPRIMET1:42;
hence 5711 is prime by NAT_4:14; :: thesis: