now :: thesis: ( not 2 divides 83 & not 3 divides 83 & not 5 divides 83 & not 7 divides 83 )

83 = (2 * 41) + 1 ;
hence not 2 divides 83 by NAT_4:9; :: thesis: ( not 3 divides 83 & not 5 divides 83 & not 7 divides 83 )
83 = (3 * 27) + 2 ;
hence not 3 divides 83 by NAT_4:9; :: thesis: ( not 5 divides 83 & not 7 divides 83 )
83 = (5 * 16) + 3 ;
hence not 5 divides 83 by NAT_4:9; :: thesis: not 7 divides 83
83 = (7 * 11) + 6 ;
hence not 7 divides 83 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 83 & n is prime holds
not n divides 83 by XPRIMET1:8;
hence 83 is prime by NAT_4:14; :: thesis: verum