now :: thesis: ( not 2 divides 431 & not 3 divides 431 & not 5 divides 431 & not 7 divides 431 & not 11 divides 431 & not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )

431 = (2 * 215) + 1 ;
hence not 2 divides 431 by NAT_4:9; :: thesis: ( not 3 divides 431 & not 5 divides 431 & not 7 divides 431 & not 11 divides 431 & not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )
431 = (3 * 143) + 2 ;
hence not 3 divides 431 by NAT_4:9; :: thesis: ( not 5 divides 431 & not 7 divides 431 & not 11 divides 431 & not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )
431 = (5 * 86) + 1 ;
hence not 5 divides 431 by NAT_4:9; :: thesis: ( not 7 divides 431 & not 11 divides 431 & not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )
431 = (7 * 61) + 4 ;
hence not 7 divides 431 by NAT_4:9; :: thesis: ( not 11 divides 431 & not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )
431 = (11 * 39) + 2 ;
hence not 11 divides 431 by NAT_4:9; :: thesis: ( not 13 divides 431 & not 17 divides 431 & not 19 divides 431 )
431 = (13 * 33) + 2 ;
hence not 13 divides 431 by NAT_4:9; :: thesis: ( not 17 divides 431 & not 19 divides 431 )
431 = (17 * 25) + 6 ;
hence not 17 divides 431 by NAT_4:9; :: thesis: not 19 divides 431
431 = (19 * 22) + 13 ;
hence not 19 divides 431 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 431 & n is prime holds
not n divides 431 by XPRIMET1:16;
hence 431 is prime by NAT_4:14; :: thesis: verum