5527 = (2 * 2763) + 1 ;
hence not 2 divides 5527 by NAT_4:9; :: thesis:
5527 = (3 * 1842) + 1 ;
hence not 3 divides 5527 by NAT_4:9; :: thesis:
5527 = (5 * 1105) + 2 ;
hence not 5 divides 5527 by NAT_4:9; :: thesis:
5527 = (7 * 789) + 4 ;
hence not 7 divides 5527 by NAT_4:9; :: thesis:
5527 = (11 * 502) + 5 ;
hence not 11 divides 5527 by NAT_4:9; :: thesis:
5527 = (13 * 425) + 2 ;
hence not 13 divides 5527 by NAT_4:9; :: thesis:
5527 = (17 * 325) + 2 ;
hence not 17 divides 5527 by NAT_4:9; :: thesis:
5527 = (19 * 290) + 17 ;
hence not 19 divides 5527 by NAT_4:9; :: thesis:
5527 = (23 * 240) + 7 ;
hence not 23 divides 5527 by NAT_4:9; :: thesis:
5527 = (29 * 190) + 17 ;
hence not 29 divides 5527 by NAT_4:9; :: thesis:
5527 = (31 * 178) + 9 ;
hence not 31 divides 5527 by NAT_4:9; :: thesis:
5527 = (37 * 149) + 14 ;
hence not 37 divides 5527 by NAT_4:9; :: thesis:
5527 = (41 * 134) + 33 ;
hence not 41 divides 5527 by NAT_4:9; :: thesis:
5527 = (43 * 128) + 23 ;
hence not 43 divides 5527 by NAT_4:9; :: thesis:
5527 = (47 * 117) + 28 ;
hence not 47 divides 5527 by NAT_4:9; :: thesis:
5527 = (53 * 104) + 15 ;
hence not 53 divides 5527 by NAT_4:9; :: thesis:
5527 = (59 * 93) + 40 ;
hence not 59 divides 5527 by NAT_4:9; :: thesis:
5527 = (61 * 90) + 37 ;
hence not 61 divides 5527 by NAT_4:9; :: thesis:
5527 = (67 * 82) + 33 ;
hence not 67 divides 5527 by NAT_4:9; :: thesis:
5527 = (71 * 77) + 60 ;
hence not 71 divides 5527 by NAT_4:9; :: thesis:
5527 = (73 * 75) + 52 ;
hence not 73 divides 5527 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5527 & n is prime holds
not n divides 5527 by XPRIMET1:42;
hence 5527 is prime by NAT_4:14; :: thesis: