reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;
reserve M for Nat;
reserve m,n for Nat;
reserve x1,x2,x3,x4 for object;
reserve e,u for object;

theorem
  for x, y being object holds
    dom <% x, y %> = {0,1} & rng <% x, y %> = {x,y}
proof
  let x,y be object;
  thus A1: dom <% x, y %> = {0,1} by Th35, CARD_1:50;
  now
    let b be object;
    hereby
      assume b in rng <% x, y %>;
      then consider a being object such that
        A2: a in dom <% x, y %> & <% x, y %>.a = b by FUNCT_1:def 3;
      per cases by A1, A2, TARSKI:def 2;
      suppose a = 0;
        hence b in {x,y} by A2, TARSKI:def 2;
      end;
      suppose a = 1;
        hence b in {x,y} by A2, TARSKI:def 2;
      end;
    end;
    assume b in {x, y};
    then per cases by TARSKI:def 2;
    suppose b = x;
      then 0 in dom <% x, y %> & <% x, y %>.0 = b
        by A1, TARSKI:def 2;
      hence b in rng <% x, y %> by FUNCT_1:3;
    end;
    suppose b = y;
      then 1 in dom <% x, y %> & <% x, y %>.1 = b
        by A1, TARSKI:def 2;
      hence b in rng <% x, y %> by FUNCT_1:3;
    end;
  end;
  hence thesis by TARSKI:2;
end;
