theorem Th64:
   for a be Element of A holds a in Non_ZeroDiv_Set(A) iff a <> 0.A
   proof
     let a be Element of A;
     thus a in Non_ZeroDiv_Set(A) implies a <> 0.A
     proof
       assume a in Non_ZeroDiv_Set(A); then
       a in [#]A \ {0.A} by Lm63; then
       a in [#]A & not a in {0.A} by XBOOLE_0:def 5;
       hence thesis by TARSKI:def 1;
     end;
     assume a <> 0.A; then
     not a in {0.A} by TARSKI:def 1; then
     a in [#]A \ {0.A} by XBOOLE_0:def 5;
     hence thesis by Lm63;
   end;
