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

theorem Conv:
  for R being non empty RelStr st
    (for X being Subset of R holds UAp ((UAp X)`) c= (UAp X)`) holds
    for X being Subset of R holds (LAp X)` c= LAp ((LAp X)`)
  proof
    let R be non empty RelStr;
    assume
TR: (for X being Subset of R holds UAp ((UAp X)`) c= (UAp X)`);
    let X be Subset of R;
H1: LAp X = Lap X by ROUGHS_2:9
         .= (UAp X`)` by ROUGHS_2:def 9; then
    (UAp X`)`` c= (UAp (LAp X))` by TR,SUBSET_1:12; then
    (LAp X)` c= (Uap (LAp X))` by H1,ROUGHS_2:8; then
    (LAp X)` c= (LAp (LAp X)`)`` by ROUGHS_2:def 8;
    hence thesis;
  end;
