6299 = (2 * 3149) + 1 ;
hence not 2 divides 6299 by NAT_4:9; :: thesis:
6299 = (3 * 2099) + 2 ;
hence not 3 divides 6299 by NAT_4:9; :: thesis:
6299 = (5 * 1259) + 4 ;
hence not 5 divides 6299 by NAT_4:9; :: thesis:
6299 = (7 * 899) + 6 ;
hence not 7 divides 6299 by NAT_4:9; :: thesis:
6299 = (11 * 572) + 7 ;
hence not 11 divides 6299 by NAT_4:9; :: thesis:
6299 = (13 * 484) + 7 ;
hence not 13 divides 6299 by NAT_4:9; :: thesis:
6299 = (17 * 370) + 9 ;
hence not 17 divides 6299 by NAT_4:9; :: thesis:
6299 = (19 * 331) + 10 ;
hence not 19 divides 6299 by NAT_4:9; :: thesis:
6299 = (23 * 273) + 20 ;
hence not 23 divides 6299 by NAT_4:9; :: thesis:
6299 = (29 * 217) + 6 ;
hence not 29 divides 6299 by NAT_4:9; :: thesis:
6299 = (31 * 203) + 6 ;
hence not 31 divides 6299 by NAT_4:9; :: thesis:
6299 = (37 * 170) + 9 ;
hence not 37 divides 6299 by NAT_4:9; :: thesis:
6299 = (41 * 153) + 26 ;
hence not 41 divides 6299 by NAT_4:9; :: thesis:
6299 = (43 * 146) + 21 ;
hence not 43 divides 6299 by NAT_4:9; :: thesis:
6299 = (47 * 134) + 1 ;
hence not 47 divides 6299 by NAT_4:9; :: thesis:
6299 = (53 * 118) + 45 ;
hence not 53 divides 6299 by NAT_4:9; :: thesis:
6299 = (59 * 106) + 45 ;
hence not 59 divides 6299 by NAT_4:9; :: thesis:
6299 = (61 * 103) + 16 ;
hence not 61 divides 6299 by NAT_4:9; :: thesis:
6299 = (67 * 94) + 1 ;
hence not 67 divides 6299 by NAT_4:9; :: thesis:
6299 = (71 * 88) + 51 ;
hence not 71 divides 6299 by NAT_4:9; :: thesis:
6299 = (73 * 86) + 21 ;
hence not 73 divides 6299 by NAT_4:9; :: thesis:
6299 = (79 * 79) + 58 ;
hence not 79 divides 6299 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6299 & n is prime holds
not n divides 6299 by XPRIMET1:44;
hence 6299 is prime by NAT_4:14; :: thesis: