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

theorem Sat7Sat:
  for R being finite non empty RelStr st
    R is satisfying(7L') holds R is satisfying(7H')
  proof
    let R be finite non empty RelStr;
    assume
tr: R is satisfying(7L');
    for X being Subset of R holds (UAp X)` c= UAp ((UAp X)`)
    proof
      let X be Subset of R;
H1:   UAp X = Uap X by ROUGHS_2:8 .= (LAp X`)` by ROUGHS_2:def 8; then
      (LAp X`)`` c= (LAp (UAp X))` by tr,SUBSET_1:12; then
      (UAp X)` c= (Lap (UAp X))` by H1,ROUGHS_2:9; then
      (UAp X)` c= (UAp (UAp X)`)`` by ROUGHS_2:def 9;
      hence thesis;
    end;
    hence thesis;
  end;
