5101 = (2 * 2550) + 1 ;
hence not 2 divides 5101 by NAT_4:9; :: thesis:
5101 = (3 * 1700) + 1 ;
hence not 3 divides 5101 by NAT_4:9; :: thesis:
5101 = (5 * 1020) + 1 ;
hence not 5 divides 5101 by NAT_4:9; :: thesis:
5101 = (7 * 728) + 5 ;
hence not 7 divides 5101 by NAT_4:9; :: thesis:
5101 = (11 * 463) + 8 ;
hence not 11 divides 5101 by NAT_4:9; :: thesis:
5101 = (13 * 392) + 5 ;
hence not 13 divides 5101 by NAT_4:9; :: thesis:
5101 = (17 * 300) + 1 ;
hence not 17 divides 5101 by NAT_4:9; :: thesis:
5101 = (19 * 268) + 9 ;
hence not 19 divides 5101 by NAT_4:9; :: thesis:
5101 = (23 * 221) + 18 ;
hence not 23 divides 5101 by NAT_4:9; :: thesis:
5101 = (29 * 175) + 26 ;
hence not 29 divides 5101 by NAT_4:9; :: thesis:
5101 = (31 * 164) + 17 ;
hence not 31 divides 5101 by NAT_4:9; :: thesis:
5101 = (37 * 137) + 32 ;
hence not 37 divides 5101 by NAT_4:9; :: thesis:
5101 = (41 * 124) + 17 ;
hence not 41 divides 5101 by NAT_4:9; :: thesis:
5101 = (43 * 118) + 27 ;
hence not 43 divides 5101 by NAT_4:9; :: thesis:
5101 = (47 * 108) + 25 ;
hence not 47 divides 5101 by NAT_4:9; :: thesis:
5101 = (53 * 96) + 13 ;
hence not 53 divides 5101 by NAT_4:9; :: thesis:
5101 = (59 * 86) + 27 ;
hence not 59 divides 5101 by NAT_4:9; :: thesis:
5101 = (61 * 83) + 38 ;
hence not 61 divides 5101 by NAT_4:9; :: thesis:
5101 = (67 * 76) + 9 ;
hence not 67 divides 5101 by NAT_4:9; :: thesis:
5101 = (71 * 71) + 60 ;
hence not 71 divides 5101 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5101 & n is prime holds
not n divides 5101 by XPRIMET1:40;
hence 5101 is prime by NAT_4:14; :: thesis: