 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem R224: :: dual version of ROUGHS_2:24
  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.X c= L.(L.X) holds
      for X being Subset of A holds U.(U.X) c= U.X
  proof
    let A be non empty set,
        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.X c= L.(L.X);
    let X be Subset of A;
A3: U.X = (L.X`)` by ROUGHS_2:def 14,A1;
    U.(U.X) = U.(L.X`)` by ROUGHS_2:def 14,A1
           .= (L.(L.X`)``)` by ROUGHS_2:def 14,A1
           .= (L.(L.X`))`;
    hence thesis by A3,A2,SUBSET_1:12;
  end;
