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

theorem Conv2:
  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;
    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.X)` c= L.((L.X)`);
    let X be Subset of A;
A3: U.X = (L.X`)` by ROUGHS_2:def 14,A1;
    (L.(U.X)``)` = U.((U.X)`) by ROUGHS_2:def 14,A1;
    hence thesis by A2,A3,SUBSET_1:12;
  end;
