reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem
  for x,y being object st x <> y holds <% x, y %> is one-to-one
proof
  let x,y be object;
  assume A1: x <> y;
  now
    let x1, x2 be object;
    assume A2: x1 in dom <%x,y%> & x2 in dom <%x,y%> & <%x,y%>.x1=<%x,y%>.x2;
    then x1 in {0,1} & x2 in {0,1} by Th7;
    then per cases by TARSKI:def 2;
    suppose x1 = 0 & x2 = 0;
      hence x1 = x2;
    end;
    suppose x1 = 0 & x2 = 1;
      hence x1 = x2 by A1, A2; :: by contradiction
    end;
    suppose x1 = 1 & x2 = 0;
      hence x1 = x2 by A1, A2; :: by contradiction
    end;
    suppose x1 = 1 & x2 = 1;
      hence x1 = x2;
    end;
  end;
  hence thesis by FUNCT_1:def 4;
end;
