7243 = (2 * 3621) + 1 ;
hence not 2 divides 7243 by NAT_4:9; :: thesis:
7243 = (3 * 2414) + 1 ;
hence not 3 divides 7243 by NAT_4:9; :: thesis:
7243 = (5 * 1448) + 3 ;
hence not 5 divides 7243 by NAT_4:9; :: thesis:
7243 = (7 * 1034) + 5 ;
hence not 7 divides 7243 by NAT_4:9; :: thesis:
7243 = (11 * 658) + 5 ;
hence not 11 divides 7243 by NAT_4:9; :: thesis:
7243 = (13 * 557) + 2 ;
hence not 13 divides 7243 by NAT_4:9; :: thesis:
7243 = (17 * 426) + 1 ;
hence not 17 divides 7243 by NAT_4:9; :: thesis:
7243 = (19 * 381) + 4 ;
hence not 19 divides 7243 by NAT_4:9; :: thesis:
7243 = (23 * 314) + 21 ;
hence not 23 divides 7243 by NAT_4:9; :: thesis:
7243 = (29 * 249) + 22 ;
hence not 29 divides 7243 by NAT_4:9; :: thesis:
7243 = (31 * 233) + 20 ;
hence not 31 divides 7243 by NAT_4:9; :: thesis:
7243 = (37 * 195) + 28 ;
hence not 37 divides 7243 by NAT_4:9; :: thesis:
7243 = (41 * 176) + 27 ;
hence not 41 divides 7243 by NAT_4:9; :: thesis:
7243 = (43 * 168) + 19 ;
hence not 43 divides 7243 by NAT_4:9; :: thesis:
7243 = (47 * 154) + 5 ;
hence not 47 divides 7243 by NAT_4:9; :: thesis:
7243 = (53 * 136) + 35 ;
hence not 53 divides 7243 by NAT_4:9; :: thesis:
7243 = (59 * 122) + 45 ;
hence not 59 divides 7243 by NAT_4:9; :: thesis:
7243 = (61 * 118) + 45 ;
hence not 61 divides 7243 by NAT_4:9; :: thesis:
7243 = (67 * 108) + 7 ;
hence not 67 divides 7243 by NAT_4:9; :: thesis:
7243 = (71 * 102) + 1 ;
hence not 71 divides 7243 by NAT_4:9; :: thesis:
7243 = (73 * 99) + 16 ;
hence not 73 divides 7243 by NAT_4:9; :: thesis:
7243 = (79 * 91) + 54 ;
hence not 79 divides 7243 by NAT_4:9; :: thesis:
7243 = (83 * 87) + 22 ;
hence not 83 divides 7243 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7243 & n is prime holds
not n divides 7243 by XPRIMET1:46;
hence 7243 is prime by NAT_4:14; :: thesis: