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

theorem Conv3:
  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 U.((U.X)`) c= (U.X)`) holds
      for X being Subset of A holds (L.X)` c= L.((L.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 U.((U.X)`) c= (U.X)`;
    let X be Subset of A;
    (U.X`)`` c= (U.((U.X`)`))` by A2,SUBSET_1:12; then
    (L.X``)` c= (U.((U.X`)`))` by A1,ROUGHS_2:def 14; then
    (L.X)` c= (U.((L.X)``))` by A1,ROUGHS_2:def 14; then
    (L.X)` c= (L.(L.X)`)`` by A1,ROUGHS_2:def 14;
    hence thesis;
  end;
