7951 = (2 * 3975) + 1 ;
hence not 2 divides 7951 by NAT_4:9; :: thesis:
7951 = (3 * 2650) + 1 ;
hence not 3 divides 7951 by NAT_4:9; :: thesis:
7951 = (5 * 1590) + 1 ;
hence not 5 divides 7951 by NAT_4:9; :: thesis:
7951 = (7 * 1135) + 6 ;
hence not 7 divides 7951 by NAT_4:9; :: thesis:
7951 = (11 * 722) + 9 ;
hence not 11 divides 7951 by NAT_4:9; :: thesis:
7951 = (13 * 611) + 8 ;
hence not 13 divides 7951 by NAT_4:9; :: thesis:
7951 = (17 * 467) + 12 ;
hence not 17 divides 7951 by NAT_4:9; :: thesis:
7951 = (19 * 418) + 9 ;
hence not 19 divides 7951 by NAT_4:9; :: thesis:
7951 = (23 * 345) + 16 ;
hence not 23 divides 7951 by NAT_4:9; :: thesis:
7951 = (29 * 274) + 5 ;
hence not 29 divides 7951 by NAT_4:9; :: thesis:
7951 = (31 * 256) + 15 ;
hence not 31 divides 7951 by NAT_4:9; :: thesis:
7951 = (37 * 214) + 33 ;
hence not 37 divides 7951 by NAT_4:9; :: thesis:
7951 = (41 * 193) + 38 ;
hence not 41 divides 7951 by NAT_4:9; :: thesis:
7951 = (43 * 184) + 39 ;
hence not 43 divides 7951 by NAT_4:9; :: thesis:
7951 = (47 * 169) + 8 ;
hence not 47 divides 7951 by NAT_4:9; :: thesis:
7951 = (53 * 150) + 1 ;
hence not 53 divides 7951 by NAT_4:9; :: thesis:
7951 = (59 * 134) + 45 ;
hence not 59 divides 7951 by NAT_4:9; :: thesis:
7951 = (61 * 130) + 21 ;
hence not 61 divides 7951 by NAT_4:9; :: thesis:
7951 = (67 * 118) + 45 ;
hence not 67 divides 7951 by NAT_4:9; :: thesis:
7951 = (71 * 111) + 70 ;
hence not 71 divides 7951 by NAT_4:9; :: thesis:
7951 = (73 * 108) + 67 ;
hence not 73 divides 7951 by NAT_4:9; :: thesis:
7951 = (79 * 100) + 51 ;
hence not 79 divides 7951 by NAT_4:9; :: thesis:
7951 = (83 * 95) + 66 ;
hence not 83 divides 7951 by NAT_4:9; :: thesis:
7951 = (89 * 89) + 30 ;
hence not 89 divides 7951 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7951 & n is prime holds
not n divides 7951 by XPRIMET1:48;
hence 7951 is prime by NAT_4:14; :: thesis: