now :: thesis: for n being Element of NAT st 1 < n & n * n <= 83 & n is prime holds
not n divides 83

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 83 & n is prime implies not n divides 83 )
83 = (2 * 41) + 1 ;
then A1: not 2 divides 83 by Th9;
83 = (3 * 27) + 2 ;
then A2: not 3 divides 83 by Th9;
83 = (13 * 6) + 5 ;
then A3: not 13 divides 83 by Th9;
83 = (11 * 7) + 6 ;
then A4: not 11 divides 83 by Th9;
83 = (19 * 4) + 7 ;
then A5: not 19 divides 83 by Th9;
83 = (17 * 4) + 15 ;
then A6: not 17 divides 83 by Th9;
83 = (23 * 3) + 14 ;
then A7: not 23 divides 83 by Th9;
83 = (7 * 11) + 6 ;
then A8: not 7 divides 83 by Th9;
83 = (5 * 16) + 3 ;
then A9: not 5 divides 83 by Th9;
assume ( 1 < n & n * n <= 83 & n is prime ) ; :: thesis: not n divides 83
hence not n divides 83 by A1, A2, A9, A8, A4, A3, A6, A5, A7, Lm6; :: thesis: verum

end;

hence 83 is prime by Th14; :: thesis: verum