now :: thesis: ( not 2 divides 2477 & not 3 divides 2477 & not 5 divides 2477 & not 7 divides 2477 & not 11 divides 2477 & not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )

2477 = (2 * 1238) + 1 ;
hence not 2 divides 2477 by NAT_4:9; :: thesis: ( not 3 divides 2477 & not 5 divides 2477 & not 7 divides 2477 & not 11 divides 2477 & not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (3 * 825) + 2 ;
hence not 3 divides 2477 by NAT_4:9; :: thesis: ( not 5 divides 2477 & not 7 divides 2477 & not 11 divides 2477 & not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (5 * 495) + 2 ;
hence not 5 divides 2477 by NAT_4:9; :: thesis: ( not 7 divides 2477 & not 11 divides 2477 & not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (7 * 353) + 6 ;
hence not 7 divides 2477 by NAT_4:9; :: thesis: ( not 11 divides 2477 & not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (11 * 225) + 2 ;
hence not 11 divides 2477 by NAT_4:9; :: thesis: ( not 13 divides 2477 & not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (13 * 190) + 7 ;
hence not 13 divides 2477 by NAT_4:9; :: thesis: ( not 17 divides 2477 & not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (17 * 145) + 12 ;
hence not 17 divides 2477 by NAT_4:9; :: thesis: ( not 19 divides 2477 & not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (19 * 130) + 7 ;
hence not 19 divides 2477 by NAT_4:9; :: thesis: ( not 23 divides 2477 & not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (23 * 107) + 16 ;
hence not 23 divides 2477 by NAT_4:9; :: thesis: ( not 29 divides 2477 & not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (29 * 85) + 12 ;
hence not 29 divides 2477 by NAT_4:9; :: thesis: ( not 31 divides 2477 & not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (31 * 79) + 28 ;
hence not 31 divides 2477 by NAT_4:9; :: thesis: ( not 37 divides 2477 & not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (37 * 66) + 35 ;
hence not 37 divides 2477 by NAT_4:9; :: thesis: ( not 41 divides 2477 & not 43 divides 2477 & not 47 divides 2477 )
2477 = (41 * 60) + 17 ;
hence not 41 divides 2477 by NAT_4:9; :: thesis: ( not 43 divides 2477 & not 47 divides 2477 )
2477 = (43 * 57) + 26 ;
hence not 43 divides 2477 by NAT_4:9; :: thesis: not 47 divides 2477
2477 = (47 * 52) + 33 ;
hence not 47 divides 2477 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2477 & n is prime holds
not n divides 2477 by XPRIMET1:30;
hence 2477 is prime by NAT_4:14; :: thesis: verum