
theorem Th24:
  for A being non empty set,
      L, U being Function of bool A, bool A st
    U = Flip L &
    for X being Subset of A holds L.(L.X) c= L.X holds
      for X being Subset of A holds U.X c= U.(U.X)
  proof
    let A be non empty set;
    let L, U be Function of bool A, bool A;
    assume that
A1: U = Flip L and
A2: for X being Subset of A holds L.(L.X) c= L.X;
    let X be Subset of A;
A3: U.X = (L.X`)` by Def14,A1;
    U.(U.X) = U.(L.X`)` by Def14,A1
           .= (L.(L.X`)``)` by Def14,A1
           .= (L.(L.X`))`;
    hence thesis by A3,A2,SUBSET_1:12;
  end;
