reserve v,x for object;
reserve D,V,A for set;
reserve n for Nat;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve d for Element of D;
reserve f,g for BinominativeFunction of D;
reserve p,q,r,s for PartialPredicate of D;

theorem
  <*p,f,q*> is SFHT of D & q ||= r & dom r c= dom q
  implies <*p,f,r*> is SFHT of D
  proof
    assume that
A1: <*p,f,q*> is SFHT of D and
A2: q ||= r and
A3: dom r c= dom q;
    for d holds d in dom p & p.d = TRUE & d in dom f & f.d in dom r implies
     r.(f.d) = TRUE
    proof
      let d;
      assume that
A4:   d in dom p and
A5:   p.d = TRUE and
A6:   d in dom f and
A7:   f.d in dom r;
      q.(f.d) = TRUE by A1,A3,A4,A5,A6,A7,Th11;
      hence r.(f.d) = TRUE by A2,A3,A7;
    end;
    then <*p,f,r*> in SFHTs(D);
    hence thesis;
  end;
