now :: thesis: ( not 2 divides 131 & not 3 divides 131 & not 5 divides 131 & not 7 divides 131 & not 11 divides 131 )

131 = (2 * 65) + 1 ;
hence not 2 divides 131 by NAT_4:9; :: thesis: ( not 3 divides 131 & not 5 divides 131 & not 7 divides 131 & not 11 divides 131 )
131 = (3 * 43) + 2 ;
hence not 3 divides 131 by NAT_4:9; :: thesis: ( not 5 divides 131 & not 7 divides 131 & not 11 divides 131 )
131 = (5 * 26) + 1 ;
hence not 5 divides 131 by NAT_4:9; :: thesis: ( not 7 divides 131 & not 11 divides 131 )
131 = (7 * 18) + 5 ;
hence not 7 divides 131 by NAT_4:9; :: thesis: not 11 divides 131
131 = (11 * 11) + 10 ;
hence not 11 divides 131 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 131 & n is prime holds
not n divides 131 by XPRIMET1:10;
hence 131 is prime by NAT_4:14; :: thesis: verum