now :: thesis: ( not 2 divides 127 & not 3 divides 127 & not 5 divides 127 & not 7 divides 127 & not 11 divides 127 )

127 = (2 * 63) + 1 ;
hence not 2 divides 127 by NAT_4:9; :: thesis: ( not 3 divides 127 & not 5 divides 127 & not 7 divides 127 & not 11 divides 127 )
127 = (3 * 42) + 1 ;
hence not 3 divides 127 by NAT_4:9; :: thesis: ( not 5 divides 127 & not 7 divides 127 & not 11 divides 127 )
127 = (5 * 25) + 2 ;
hence not 5 divides 127 by NAT_4:9; :: thesis: ( not 7 divides 127 & not 11 divides 127 )
127 = (7 * 18) + 1 ;
hence not 7 divides 127 by NAT_4:9; :: thesis: not 11 divides 127
127 = (11 * 11) + 6 ;
hence not 11 divides 127 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 127 & n is prime holds
not n divides 127 by XPRIMET1:10;
hence 127 is prime by NAT_4:14; :: thesis: verum