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 Th15:
  <*p,f,q*> is SFHT of D & r ||= p implies <*r,f,q*> is SFHT of D
  proof
    assume that
A1: <*p,f,q*> is SFHT of D and
A2: r ||= p;
    for d holds d in dom r & r.d = TRUE & d in dom f & f.d in dom q implies
     q.(f.d) = TRUE
    proof
      let d;
      assume d in dom r & r.d = TRUE;
      then d in dom p & p.d = TRUE by A2;
      hence thesis by A1,Th11;
    end;
    then <*r,f,q*> in SFHTs(D);
    hence thesis;
  end;
