5563 = (2 * 2781) + 1 ;
hence not 2 divides 5563 by NAT_4:9; :: thesis:
5563 = (3 * 1854) + 1 ;
hence not 3 divides 5563 by NAT_4:9; :: thesis:
5563 = (5 * 1112) + 3 ;
hence not 5 divides 5563 by NAT_4:9; :: thesis:
5563 = (7 * 794) + 5 ;
hence not 7 divides 5563 by NAT_4:9; :: thesis:
5563 = (11 * 505) + 8 ;
hence not 11 divides 5563 by NAT_4:9; :: thesis:
5563 = (13 * 427) + 12 ;
hence not 13 divides 5563 by NAT_4:9; :: thesis:
5563 = (17 * 327) + 4 ;
hence not 17 divides 5563 by NAT_4:9; :: thesis:
5563 = (19 * 292) + 15 ;
hence not 19 divides 5563 by NAT_4:9; :: thesis:
5563 = (23 * 241) + 20 ;
hence not 23 divides 5563 by NAT_4:9; :: thesis:
5563 = (29 * 191) + 24 ;
hence not 29 divides 5563 by NAT_4:9; :: thesis:
5563 = (31 * 179) + 14 ;
hence not 31 divides 5563 by NAT_4:9; :: thesis:
5563 = (37 * 150) + 13 ;
hence not 37 divides 5563 by NAT_4:9; :: thesis:
5563 = (41 * 135) + 28 ;
hence not 41 divides 5563 by NAT_4:9; :: thesis:
5563 = (43 * 129) + 16 ;
hence not 43 divides 5563 by NAT_4:9; :: thesis:
5563 = (47 * 118) + 17 ;
hence not 47 divides 5563 by NAT_4:9; :: thesis:
5563 = (53 * 104) + 51 ;
hence not 53 divides 5563 by NAT_4:9; :: thesis:
5563 = (59 * 94) + 17 ;
hence not 59 divides 5563 by NAT_4:9; :: thesis:
5563 = (61 * 91) + 12 ;
hence not 61 divides 5563 by NAT_4:9; :: thesis:
5563 = (67 * 83) + 2 ;
hence not 67 divides 5563 by NAT_4:9; :: thesis:
5563 = (71 * 78) + 25 ;
hence not 71 divides 5563 by NAT_4:9; :: thesis:
5563 = (73 * 76) + 15 ;
hence not 73 divides 5563 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5563 & n is prime holds
not n divides 5563 by XPRIMET1:42;
hence 5563 is prime by NAT_4:14; :: thesis: