
theorem
for R being Ring holds Char R = 1 iff R is degenerated
proof
let R be Ring;
hereby assume Char R = 1;
   then 1 '*' 1.R = 0.R & 1 <> 0 &
        for m being positive Nat st m < 1 holds m '*' 1.R <> 0.R
     by RING_3:def 5;
   hence R is degenerated by RING_3:60;
   end;
assume R is degenerated;
   then A: 1 '*' 1.R = 0.R by RING_3:60;
   for m being positive Nat st m < 1 holds m'*'1.R <> 0.R by NAT_1:14;
   hence Char R = 1 by A,RING_3:def 5;
end;
