5501 = (2 * 2750) + 1 ;
hence not 2 divides 5501 by NAT_4:9; :: thesis:
5501 = (3 * 1833) + 2 ;
hence not 3 divides 5501 by NAT_4:9; :: thesis:
5501 = (5 * 1100) + 1 ;
hence not 5 divides 5501 by NAT_4:9; :: thesis:
5501 = (7 * 785) + 6 ;
hence not 7 divides 5501 by NAT_4:9; :: thesis:
5501 = (11 * 500) + 1 ;
hence not 11 divides 5501 by NAT_4:9; :: thesis:
5501 = (13 * 423) + 2 ;
hence not 13 divides 5501 by NAT_4:9; :: thesis:
5501 = (17 * 323) + 10 ;
hence not 17 divides 5501 by NAT_4:9; :: thesis:
5501 = (19 * 289) + 10 ;
hence not 19 divides 5501 by NAT_4:9; :: thesis:
5501 = (23 * 239) + 4 ;
hence not 23 divides 5501 by NAT_4:9; :: thesis:
5501 = (29 * 189) + 20 ;
hence not 29 divides 5501 by NAT_4:9; :: thesis:
5501 = (31 * 177) + 14 ;
hence not 31 divides 5501 by NAT_4:9; :: thesis:
5501 = (37 * 148) + 25 ;
hence not 37 divides 5501 by NAT_4:9; :: thesis:
5501 = (41 * 134) + 7 ;
hence not 41 divides 5501 by NAT_4:9; :: thesis:
5501 = (43 * 127) + 40 ;
hence not 43 divides 5501 by NAT_4:9; :: thesis:
5501 = (47 * 117) + 2 ;
hence not 47 divides 5501 by NAT_4:9; :: thesis:
5501 = (53 * 103) + 42 ;
hence not 53 divides 5501 by NAT_4:9; :: thesis:
5501 = (59 * 93) + 14 ;
hence not 59 divides 5501 by NAT_4:9; :: thesis:
5501 = (61 * 90) + 11 ;
hence not 61 divides 5501 by NAT_4:9; :: thesis:
5501 = (67 * 82) + 7 ;
hence not 67 divides 5501 by NAT_4:9; :: thesis:
5501 = (71 * 77) + 34 ;
hence not 71 divides 5501 by NAT_4:9; :: thesis:
5501 = (73 * 75) + 26 ;
hence not 73 divides 5501 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5501 & n is prime holds
not n divides 5501 by XPRIMET1:42;
hence 5501 is prime by NAT_4:14; :: thesis: