now :: thesis: for n being Element of NAT st 1 < n & n * n <= 19 & n is prime holds
not n divides 19

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 19 & n is prime implies not n divides 19 )
19 = (2 * 9) + 1 ;
then A1: not 2 divides 19 by Th9;
19 = (3 * 6) + 1 ;
then A2: not 3 divides 19 by Th9;
assume ( 1 < n & n * n <= 19 & n is prime ) ; :: thesis: not n divides 19
hence not n divides 19 by A1, A2, Lm3; :: thesis: verum

end;

hence 19 is prime by Th14; :: thesis: verum