7349 = (2 * 3674) + 1 ;
hence not 2 divides 7349 by NAT_4:9; :: thesis:
7349 = (3 * 2449) + 2 ;
hence not 3 divides 7349 by NAT_4:9; :: thesis:
7349 = (5 * 1469) + 4 ;
hence not 5 divides 7349 by NAT_4:9; :: thesis:
7349 = (7 * 1049) + 6 ;
hence not 7 divides 7349 by NAT_4:9; :: thesis:
7349 = (11 * 668) + 1 ;
hence not 11 divides 7349 by NAT_4:9; :: thesis:
7349 = (13 * 565) + 4 ;
hence not 13 divides 7349 by NAT_4:9; :: thesis:
7349 = (17 * 432) + 5 ;
hence not 17 divides 7349 by NAT_4:9; :: thesis:
7349 = (19 * 386) + 15 ;
hence not 19 divides 7349 by NAT_4:9; :: thesis:
7349 = (23 * 319) + 12 ;
hence not 23 divides 7349 by NAT_4:9; :: thesis:
7349 = (29 * 253) + 12 ;
hence not 29 divides 7349 by NAT_4:9; :: thesis:
7349 = (31 * 237) + 2 ;
hence not 31 divides 7349 by NAT_4:9; :: thesis:
7349 = (37 * 198) + 23 ;
hence not 37 divides 7349 by NAT_4:9; :: thesis:
7349 = (41 * 179) + 10 ;
hence not 41 divides 7349 by NAT_4:9; :: thesis:
7349 = (43 * 170) + 39 ;
hence not 43 divides 7349 by NAT_4:9; :: thesis:
7349 = (47 * 156) + 17 ;
hence not 47 divides 7349 by NAT_4:9; :: thesis:
7349 = (53 * 138) + 35 ;
hence not 53 divides 7349 by NAT_4:9; :: thesis:
7349 = (59 * 124) + 33 ;
hence not 59 divides 7349 by NAT_4:9; :: thesis:
7349 = (61 * 120) + 29 ;
hence not 61 divides 7349 by NAT_4:9; :: thesis:
7349 = (67 * 109) + 46 ;
hence not 67 divides 7349 by NAT_4:9; :: thesis:
7349 = (71 * 103) + 36 ;
hence not 71 divides 7349 by NAT_4:9; :: thesis:
7349 = (73 * 100) + 49 ;
hence not 73 divides 7349 by NAT_4:9; :: thesis:
7349 = (79 * 93) + 2 ;
hence not 79 divides 7349 by NAT_4:9; :: thesis:
7349 = (83 * 88) + 45 ;
hence not 83 divides 7349 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7349 & n is prime holds
not n divides 7349 by XPRIMET1:46;
hence 7349 is prime by NAT_4:14; :: thesis: