reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th77:
for R being Ring holds Char R = 0 iff CharSet R = {}
proof
let R be Ring;
A1: now assume A2: Char R = 0;
   now let x be object;
     assume x in CharSet R;
     then ex n being positive Nat st x = n & n '*' 1.R = 0.R;
     hence contradiction by A2,Def5;
     end;
   hence CharSet R = {} by XBOOLE_0:def 1;
   end;
now assume A3: CharSet R = {};
  now assume ex m being positive Nat st m '*' 1.R = 0.R;
    then consider m being positive Nat such that
    A4: m '*' 1.R = 0.R;
    m in {k where k is positive Nat : k '*' 1.R = 0.R} by A4;
    hence contradiction by A3;
    end;
  hence Char R = 0 by Def5;
  end;
hence thesis by A1;
end;
