
theorem Th8:
  for X being non empty set, x being Element of X holds support UnitBag x = {x}
proof
  let X be non empty set, x be Element of X;
  now
    let a be object;
    hereby
      assume
A1:   a in support UnitBag x;
      now
        assume a <> x;
        then ((EmptyBag X)+*(x, 1)).a = (EmptyBag X).a by FUNCT_7:32
          .= 0 by PBOOLE:5;
        hence contradiction by A1,PRE_POLY:def 7;
      end;
      hence a in {x} by TARSKI:def 1;
    end;
    EmptyBag X = X --> 0 by PBOOLE:def 3;
    then
A2: dom (EmptyBag X) = X;
    assume a in {x};
    then a = x by TARSKI:def 1;
    then (UnitBag x).a = 1 by A2,FUNCT_7:31;
    hence a in support UnitBag x by PRE_POLY:def 7;
  end;
  hence thesis by TARSKI:2;
end;
