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

theorem Conv4:
  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;
    (L.X`)`` c= (L.((L.X`)`))` by A2,SUBSET_1:12; then
    (U.X``)` c= (L.((L.X`)`))` by A1,ROUGHS_2:def 14; then
    (U.X)` c= (L.((U.X)``))` by A1,ROUGHS_2:def 14;
    hence thesis by A1,ROUGHS_2:def 14;
  end;
