7949 = (2 * 3974) + 1 ;
hence not 2 divides 7949 by NAT_4:9; :: thesis:
7949 = (3 * 2649) + 2 ;
hence not 3 divides 7949 by NAT_4:9; :: thesis:
7949 = (5 * 1589) + 4 ;
hence not 5 divides 7949 by NAT_4:9; :: thesis:
7949 = (7 * 1135) + 4 ;
hence not 7 divides 7949 by NAT_4:9; :: thesis:
7949 = (11 * 722) + 7 ;
hence not 11 divides 7949 by NAT_4:9; :: thesis:
7949 = (13 * 611) + 6 ;
hence not 13 divides 7949 by NAT_4:9; :: thesis:
7949 = (17 * 467) + 10 ;
hence not 17 divides 7949 by NAT_4:9; :: thesis:
7949 = (19 * 418) + 7 ;
hence not 19 divides 7949 by NAT_4:9; :: thesis:
7949 = (23 * 345) + 14 ;
hence not 23 divides 7949 by NAT_4:9; :: thesis:
7949 = (29 * 274) + 3 ;
hence not 29 divides 7949 by NAT_4:9; :: thesis:
7949 = (31 * 256) + 13 ;
hence not 31 divides 7949 by NAT_4:9; :: thesis:
7949 = (37 * 214) + 31 ;
hence not 37 divides 7949 by NAT_4:9; :: thesis:
7949 = (41 * 193) + 36 ;
hence not 41 divides 7949 by NAT_4:9; :: thesis:
7949 = (43 * 184) + 37 ;
hence not 43 divides 7949 by NAT_4:9; :: thesis:
7949 = (47 * 169) + 6 ;
hence not 47 divides 7949 by NAT_4:9; :: thesis:
7949 = (53 * 149) + 52 ;
hence not 53 divides 7949 by NAT_4:9; :: thesis:
7949 = (59 * 134) + 43 ;
hence not 59 divides 7949 by NAT_4:9; :: thesis:
7949 = (61 * 130) + 19 ;
hence not 61 divides 7949 by NAT_4:9; :: thesis:
7949 = (67 * 118) + 43 ;
hence not 67 divides 7949 by NAT_4:9; :: thesis:
7949 = (71 * 111) + 68 ;
hence not 71 divides 7949 by NAT_4:9; :: thesis:
7949 = (73 * 108) + 65 ;
hence not 73 divides 7949 by NAT_4:9; :: thesis:
7949 = (79 * 100) + 49 ;
hence not 79 divides 7949 by NAT_4:9; :: thesis:
7949 = (83 * 95) + 64 ;
hence not 83 divides 7949 by NAT_4:9; :: thesis:
7949 = (89 * 89) + 28 ;
hence not 89 divides 7949 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7949 & n is prime holds
not n divides 7949 by XPRIMET1:48;
hence 7949 is prime by NAT_4:14; :: thesis: