now :: thesis: ( not 2 divides 4177 & not 3 divides 4177 & not 5 divides 4177 & not 7 divides 4177 & not 11 divides 4177 & not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )

4177 = (2 * 2088) + 1 ;
hence not 2 divides 4177 by NAT_4:9; :: thesis: ( not 3 divides 4177 & not 5 divides 4177 & not 7 divides 4177 & not 11 divides 4177 & not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (3 * 1392) + 1 ;
hence not 3 divides 4177 by NAT_4:9; :: thesis: ( not 5 divides 4177 & not 7 divides 4177 & not 11 divides 4177 & not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (5 * 835) + 2 ;
hence not 5 divides 4177 by NAT_4:9; :: thesis: ( not 7 divides 4177 & not 11 divides 4177 & not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (7 * 596) + 5 ;
hence not 7 divides 4177 by NAT_4:9; :: thesis: ( not 11 divides 4177 & not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (11 * 379) + 8 ;
hence not 11 divides 4177 by NAT_4:9; :: thesis: ( not 13 divides 4177 & not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (13 * 321) + 4 ;
hence not 13 divides 4177 by NAT_4:9; :: thesis: ( not 17 divides 4177 & not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (17 * 245) + 12 ;
hence not 17 divides 4177 by NAT_4:9; :: thesis: ( not 19 divides 4177 & not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (19 * 219) + 16 ;
hence not 19 divides 4177 by NAT_4:9; :: thesis: ( not 23 divides 4177 & not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (23 * 181) + 14 ;
hence not 23 divides 4177 by NAT_4:9; :: thesis: ( not 29 divides 4177 & not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (29 * 144) + 1 ;
hence not 29 divides 4177 by NAT_4:9; :: thesis: ( not 31 divides 4177 & not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (31 * 134) + 23 ;
hence not 31 divides 4177 by NAT_4:9; :: thesis: ( not 37 divides 4177 & not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (37 * 112) + 33 ;
hence not 37 divides 4177 by NAT_4:9; :: thesis: ( not 41 divides 4177 & not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (41 * 101) + 36 ;
hence not 41 divides 4177 by NAT_4:9; :: thesis: ( not 43 divides 4177 & not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (43 * 97) + 6 ;
hence not 43 divides 4177 by NAT_4:9; :: thesis: ( not 47 divides 4177 & not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (47 * 88) + 41 ;
hence not 47 divides 4177 by NAT_4:9; :: thesis: ( not 53 divides 4177 & not 59 divides 4177 & not 61 divides 4177 )
4177 = (53 * 78) + 43 ;
hence not 53 divides 4177 by NAT_4:9; :: thesis: ( not 59 divides 4177 & not 61 divides 4177 )
4177 = (59 * 70) + 47 ;
hence not 59 divides 4177 by NAT_4:9; :: thesis: not 61 divides 4177
4177 = (61 * 68) + 29 ;
hence not 61 divides 4177 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 4177 & n is prime holds
not n divides 4177 by XPRIMET1:36;
hence 4177 is prime by NAT_4:14; :: thesis: verum