7603 = (2 * 3801) + 1 ;
hence not 2 divides 7603 by NAT_4:9; :: thesis:
7603 = (3 * 2534) + 1 ;
hence not 3 divides 7603 by NAT_4:9; :: thesis:
7603 = (5 * 1520) + 3 ;
hence not 5 divides 7603 by NAT_4:9; :: thesis:
7603 = (7 * 1086) + 1 ;
hence not 7 divides 7603 by NAT_4:9; :: thesis:
7603 = (11 * 691) + 2 ;
hence not 11 divides 7603 by NAT_4:9; :: thesis:
7603 = (13 * 584) + 11 ;
hence not 13 divides 7603 by NAT_4:9; :: thesis:
7603 = (17 * 447) + 4 ;
hence not 17 divides 7603 by NAT_4:9; :: thesis:
7603 = (19 * 400) + 3 ;
hence not 19 divides 7603 by NAT_4:9; :: thesis:
7603 = (23 * 330) + 13 ;
hence not 23 divides 7603 by NAT_4:9; :: thesis:
7603 = (29 * 262) + 5 ;
hence not 29 divides 7603 by NAT_4:9; :: thesis:
7603 = (31 * 245) + 8 ;
hence not 31 divides 7603 by NAT_4:9; :: thesis:
7603 = (37 * 205) + 18 ;
hence not 37 divides 7603 by NAT_4:9; :: thesis:
7603 = (41 * 185) + 18 ;
hence not 41 divides 7603 by NAT_4:9; :: thesis:
7603 = (43 * 176) + 35 ;
hence not 43 divides 7603 by NAT_4:9; :: thesis:
7603 = (47 * 161) + 36 ;
hence not 47 divides 7603 by NAT_4:9; :: thesis:
7603 = (53 * 143) + 24 ;
hence not 53 divides 7603 by NAT_4:9; :: thesis:
7603 = (59 * 128) + 51 ;
hence not 59 divides 7603 by NAT_4:9; :: thesis:
7603 = (61 * 124) + 39 ;
hence not 61 divides 7603 by NAT_4:9; :: thesis:
7603 = (67 * 113) + 32 ;
hence not 67 divides 7603 by NAT_4:9; :: thesis:
7603 = (71 * 107) + 6 ;
hence not 71 divides 7603 by NAT_4:9; :: thesis:
7603 = (73 * 104) + 11 ;
hence not 73 divides 7603 by NAT_4:9; :: thesis:
7603 = (79 * 96) + 19 ;
hence not 79 divides 7603 by NAT_4:9; :: thesis:
7603 = (83 * 91) + 50 ;
hence not 83 divides 7603 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7603 & n is prime holds
not n divides 7603 by XPRIMET1:46;
hence 7603 is prime by NAT_4:14; :: thesis: