now :: thesis: ( not 2 divides 5077 & not 3 divides 5077 & not 5 divides 5077 & not 7 divides 5077 & not 11 divides 5077 & not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )

5077 = (2 * 2538) + 1 ;
hence not 2 divides 5077 by NAT_4:9; :: thesis: ( not 3 divides 5077 & not 5 divides 5077 & not 7 divides 5077 & not 11 divides 5077 & not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (3 * 1692) + 1 ;
hence not 3 divides 5077 by NAT_4:9; :: thesis: ( not 5 divides 5077 & not 7 divides 5077 & not 11 divides 5077 & not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (5 * 1015) + 2 ;
hence not 5 divides 5077 by NAT_4:9; :: thesis: ( not 7 divides 5077 & not 11 divides 5077 & not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (7 * 725) + 2 ;
hence not 7 divides 5077 by NAT_4:9; :: thesis: ( not 11 divides 5077 & not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (11 * 461) + 6 ;
hence not 11 divides 5077 by NAT_4:9; :: thesis: ( not 13 divides 5077 & not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (13 * 390) + 7 ;
hence not 13 divides 5077 by NAT_4:9; :: thesis: ( not 17 divides 5077 & not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (17 * 298) + 11 ;
hence not 17 divides 5077 by NAT_4:9; :: thesis: ( not 19 divides 5077 & not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (19 * 267) + 4 ;
hence not 19 divides 5077 by NAT_4:9; :: thesis: ( not 23 divides 5077 & not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (23 * 220) + 17 ;
hence not 23 divides 5077 by NAT_4:9; :: thesis: ( not 29 divides 5077 & not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (29 * 175) + 2 ;
hence not 29 divides 5077 by NAT_4:9; :: thesis: ( not 31 divides 5077 & not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (31 * 163) + 24 ;
hence not 31 divides 5077 by NAT_4:9; :: thesis: ( not 37 divides 5077 & not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (37 * 137) + 8 ;
hence not 37 divides 5077 by NAT_4:9; :: thesis: ( not 41 divides 5077 & not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (41 * 123) + 34 ;
hence not 41 divides 5077 by NAT_4:9; :: thesis: ( not 43 divides 5077 & not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (43 * 118) + 3 ;
hence not 43 divides 5077 by NAT_4:9; :: thesis: ( not 47 divides 5077 & not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (47 * 108) + 1 ;
hence not 47 divides 5077 by NAT_4:9; :: thesis: ( not 53 divides 5077 & not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (53 * 95) + 42 ;
hence not 53 divides 5077 by NAT_4:9; :: thesis: ( not 59 divides 5077 & not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (59 * 86) + 3 ;
hence not 59 divides 5077 by NAT_4:9; :: thesis: ( not 61 divides 5077 & not 67 divides 5077 & not 71 divides 5077 )
5077 = (61 * 83) + 14 ;
hence not 61 divides 5077 by NAT_4:9; :: thesis: ( not 67 divides 5077 & not 71 divides 5077 )
5077 = (67 * 75) + 52 ;
hence not 67 divides 5077 by NAT_4:9; :: thesis: not 71 divides 5077
5077 = (71 * 71) + 36 ;
hence not 71 divides 5077 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 5077 & n is prime holds
not n divides 5077 by XPRIMET1:40;
hence 5077 is prime by NAT_4:14; :: thesis: verum