
theorem
  for R being non empty RelStr
  for A being Subset-Family of R, F being Subset of R
  st A = {(uparrow x)` where x is Element of R: x in F}
  holds Intersect A = (uparrow F)`
proof
  let R be non empty RelStr;
  deffunc F(Element of R) = uparrow $1;
  let A be Subset-Family of R, F be Subset of R such that
A1: A = {F(x)` where x is Element of R: x in F};
A2: COMPLEMENT A = {F(x) where x is Element of R: x in F}
  from FraenkelComplement2(A1);
  reconsider C = COMPLEMENT A as Subset-Family of R;
  COMPLEMENT C = A;
  hence Intersect A = (union C)` by YELLOW_8:6
    .= (uparrow F)` by A2,Th4;
end;
