
theorem Th19:
  for X, Y being compact non empty TopSpace, R being Subset-Family
of X, F being Subset-Family of [:Y, X:] st F is Cover of [:Y, X:] & F is open &
R = { Q where Q is open Subset of X : ex FQ being Subset-Family of [:Y, X:] st
  FQ c= F & FQ is finite & [:[#]Y, Q:] c= union FQ } holds ex C being
  Subset-Family of X st C c= R & C is finite & C is Cover of X
proof
  let X, Y be compact non empty TopSpace, R be Subset-Family of X, F be
  Subset-Family of [:Y, X:];
  assume F is Cover of [:Y, X:] & F is open & R = { Q where Q is open Subset
of X : ex FQ being Subset-Family of [:Y, X:] st FQ c= F & FQ is finite & [:[#]Y
  , Q:] c= union FQ };
  then R is open & R is Cover of X by Th18;
  then
  ex G being Subset-Family of X st G c= R & G is Cover of X & G is finite
  by COMPTS_1:def 1;
  hence thesis;
end;
