reserve T for TopSpace;

theorem Th19:
  for A being Subset of T, F being Subset-Family of T st F = { A }
  holds Int F = { Int A }
proof
  let A be Subset of T, F be Subset-Family of T;
  reconsider C = Int F as set;
  assume
A1: F = { A };
  for B being object holds B in C iff B = Int A
  proof
    let B be object;
A2: now
      assume
A3:   B = Int A;
      ex M being Subset of T st B = Int M & M in F
      proof
        take A;
        thus thesis by A1,A3,TARSKI:def 1;
      end;
      hence B in C by Def1;
    end;
    now
      assume
A4:   B in C;
      then reconsider B0 = B as Subset of T;
      ex M being Subset of T st B0 = Int M & M in F by A4,Def1;
      hence B = Int A by A1,TARSKI:def 1;
    end;
    hence thesis by A2;
  end;
  hence thesis by TARSKI:def 1;
end;
