reserve r for Real;

theorem
  for x being set holds Empty^2-to-zero.(x,x) = 0
proof
  let x be set;
  per cases;
  suppose [x,x] in [:1,1:];
    hence thesis by CARD_1:49,FUNCT_2:50;
  end;
  suppose not [x,x] in [:1,1:]; then
    not [x,x] in dom Empty^2-to-zero;
    hence thesis by FUNCT_1:def 2;
  end;
end;
