
theorem
  for T being non empty TopSpace, x,y being Element of T for V being
  open Subset of T st x in V & not y in V holds chi(V, the carrier of T)*
   ((0,1) --> (y,x)) = id Sierpinski_Space
proof
  let T be non empty TopSpace;
  let x,y be Element of T, V be open Subset of T such that
A1: x in V and
A2: not y in V;
  reconsider c = chi(V, the carrier of T) as Function of T, Sierpinski_Space
  by Th45;
A3: c.x = 1 by A1,FUNCT_3:def 3;
A4: the carrier of Sierpinski_Space = {0,{0}} by WAYBEL18:def 9, CARD_1:49;
  then reconsider i = (0,{0}) --> (y,x) as Function of Sierpinski_Space, T;
A5: i.1 = x by FUNCT_4:63, CARD_1:49;
A6: c.y = 0 by A2,FUNCT_3:def 3;
A7: i.0 = y by FUNCT_4:63;
  now
    thus c*i is Function of Sierpinski_Space, Sierpinski_Space;
    let a be Element of Sierpinski_Space;
    a = 0 or a = 1 by A4,TARSKI:def 2, CARD_1:49;
    hence (c*i).a = a by A7,A5,A3,A6,FUNCT_2:15
      .= (id Sierpinski_Space).a;
  end;
  hence thesis by FUNCT_2:63, CARD_1:49;
end;
