7417 = (2 * 3708) + 1 ;
hence not 2 divides 7417 by NAT_4:9; :: thesis:
7417 = (3 * 2472) + 1 ;
hence not 3 divides 7417 by NAT_4:9; :: thesis:
7417 = (5 * 1483) + 2 ;
hence not 5 divides 7417 by NAT_4:9; :: thesis:
7417 = (7 * 1059) + 4 ;
hence not 7 divides 7417 by NAT_4:9; :: thesis:
7417 = (11 * 674) + 3 ;
hence not 11 divides 7417 by NAT_4:9; :: thesis:
7417 = (13 * 570) + 7 ;
hence not 13 divides 7417 by NAT_4:9; :: thesis:
7417 = (17 * 436) + 5 ;
hence not 17 divides 7417 by NAT_4:9; :: thesis:
7417 = (19 * 390) + 7 ;
hence not 19 divides 7417 by NAT_4:9; :: thesis:
7417 = (23 * 322) + 11 ;
hence not 23 divides 7417 by NAT_4:9; :: thesis:
7417 = (29 * 255) + 22 ;
hence not 29 divides 7417 by NAT_4:9; :: thesis:
7417 = (31 * 239) + 8 ;
hence not 31 divides 7417 by NAT_4:9; :: thesis:
7417 = (37 * 200) + 17 ;
hence not 37 divides 7417 by NAT_4:9; :: thesis:
7417 = (41 * 180) + 37 ;
hence not 41 divides 7417 by NAT_4:9; :: thesis:
7417 = (43 * 172) + 21 ;
hence not 43 divides 7417 by NAT_4:9; :: thesis:
7417 = (47 * 157) + 38 ;
hence not 47 divides 7417 by NAT_4:9; :: thesis:
7417 = (53 * 139) + 50 ;
hence not 53 divides 7417 by NAT_4:9; :: thesis:
7417 = (59 * 125) + 42 ;
hence not 59 divides 7417 by NAT_4:9; :: thesis:
7417 = (61 * 121) + 36 ;
hence not 61 divides 7417 by NAT_4:9; :: thesis:
7417 = (67 * 110) + 47 ;
hence not 67 divides 7417 by NAT_4:9; :: thesis:
7417 = (71 * 104) + 33 ;
hence not 71 divides 7417 by NAT_4:9; :: thesis:
7417 = (73 * 101) + 44 ;
hence not 73 divides 7417 by NAT_4:9; :: thesis:
7417 = (79 * 93) + 70 ;
hence not 79 divides 7417 by NAT_4:9; :: thesis:
7417 = (83 * 89) + 30 ;
hence not 83 divides 7417 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7417 & n is prime holds
not n divides 7417 by XPRIMET1:46;
hence 7417 is prime by NAT_4:14; :: thesis: