now :: thesis: ( not 2 divides 947 & not 3 divides 947 & not 5 divides 947 & not 7 divides 947 & not 11 divides 947 & not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )

947 = (2 * 473) + 1 ;
hence not 2 divides 947 by NAT_4:9; :: thesis: ( not 3 divides 947 & not 5 divides 947 & not 7 divides 947 & not 11 divides 947 & not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (3 * 315) + 2 ;
hence not 3 divides 947 by NAT_4:9; :: thesis: ( not 5 divides 947 & not 7 divides 947 & not 11 divides 947 & not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (5 * 189) + 2 ;
hence not 5 divides 947 by NAT_4:9; :: thesis: ( not 7 divides 947 & not 11 divides 947 & not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (7 * 135) + 2 ;
hence not 7 divides 947 by NAT_4:9; :: thesis: ( not 11 divides 947 & not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (11 * 86) + 1 ;
hence not 11 divides 947 by NAT_4:9; :: thesis: ( not 13 divides 947 & not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (13 * 72) + 11 ;
hence not 13 divides 947 by NAT_4:9; :: thesis: ( not 17 divides 947 & not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (17 * 55) + 12 ;
hence not 17 divides 947 by NAT_4:9; :: thesis: ( not 19 divides 947 & not 23 divides 947 & not 29 divides 947 )
947 = (19 * 49) + 16 ;
hence not 19 divides 947 by NAT_4:9; :: thesis: ( not 23 divides 947 & not 29 divides 947 )
947 = (23 * 41) + 4 ;
hence not 23 divides 947 by NAT_4:9; :: thesis: not 29 divides 947
947 = (29 * 32) + 19 ;
hence not 29 divides 947 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 947 & n is prime holds
not n divides 947 by XPRIMET1:20;
hence 947 is prime by NAT_4:14; :: thesis: verum