6563 = (2 * 3281) + 1 ;
hence not 2 divides 6563 by NAT_4:9; :: thesis:
6563 = (3 * 2187) + 2 ;
hence not 3 divides 6563 by NAT_4:9; :: thesis:
6563 = (5 * 1312) + 3 ;
hence not 5 divides 6563 by NAT_4:9; :: thesis:
6563 = (7 * 937) + 4 ;
hence not 7 divides 6563 by NAT_4:9; :: thesis:
6563 = (11 * 596) + 7 ;
hence not 11 divides 6563 by NAT_4:9; :: thesis:
6563 = (13 * 504) + 11 ;
hence not 13 divides 6563 by NAT_4:9; :: thesis:
6563 = (17 * 386) + 1 ;
hence not 17 divides 6563 by NAT_4:9; :: thesis:
6563 = (19 * 345) + 8 ;
hence not 19 divides 6563 by NAT_4:9; :: thesis:
6563 = (23 * 285) + 8 ;
hence not 23 divides 6563 by NAT_4:9; :: thesis:
6563 = (29 * 226) + 9 ;
hence not 29 divides 6563 by NAT_4:9; :: thesis:
6563 = (31 * 211) + 22 ;
hence not 31 divides 6563 by NAT_4:9; :: thesis:
6563 = (37 * 177) + 14 ;
hence not 37 divides 6563 by NAT_4:9; :: thesis:
6563 = (41 * 160) + 3 ;
hence not 41 divides 6563 by NAT_4:9; :: thesis:
6563 = (43 * 152) + 27 ;
hence not 43 divides 6563 by NAT_4:9; :: thesis:
6563 = (47 * 139) + 30 ;
hence not 47 divides 6563 by NAT_4:9; :: thesis:
6563 = (53 * 123) + 44 ;
hence not 53 divides 6563 by NAT_4:9; :: thesis:
6563 = (59 * 111) + 14 ;
hence not 59 divides 6563 by NAT_4:9; :: thesis:
6563 = (61 * 107) + 36 ;
hence not 61 divides 6563 by NAT_4:9; :: thesis:
6563 = (67 * 97) + 64 ;
hence not 67 divides 6563 by NAT_4:9; :: thesis:
6563 = (71 * 92) + 31 ;
hence not 71 divides 6563 by NAT_4:9; :: thesis:
6563 = (73 * 89) + 66 ;
hence not 73 divides 6563 by NAT_4:9; :: thesis:
6563 = (79 * 83) + 6 ;
hence not 79 divides 6563 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6563 & n is prime holds
not n divides 6563 by XPRIMET1:44;
hence 6563 is prime by NAT_4:14; :: thesis: