reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th13:
  for X, Y being TopSpace, A being Subset of [:X,Y:] st A is open
  holds A = union Base-Appr A
proof
  let X, Y be TopSpace, A be Subset of [:X,Y:];
  assume A is open;
  then consider B being Subset-Family of [:X,Y:] such that
A1: A = union B and
A2: for e st e in B ex X1 being Subset of X, Y1 being Subset of Y st e =
  [:X1,Y1:] & X1 is open & Y1 is open by Th5;
  thus A c= union Base-Appr A
  proof
    let e be object;
    assume e in A;
    then consider u such that
A3: e in u and
A4: u in B by A1,TARSKI:def 4;
    ( ex X1 being Subset of X, Y1 being Subset of Y st u = [:X1,Y1:] & X1
    is open & Y1 is open)& u c= A by A1,A2,A4,ZFMISC_1:74;
    then u in Base-Appr A;
    hence thesis by A3,TARSKI:def 4;
  end;
  thus thesis by Th12;
end;
