reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;

theorem Th27:
  for o be OperSymbol of S1 for A be OSSubset of OU0 for B be
OSSubset of OU0 st B in OSSubSort(A) holds rng (Den(o,OU0)|(((OSMSubSort A)# *
  (the Arity of S1)).o)) c= (B * (the ResultSort of S1)).o
proof
  let o be OperSymbol of S1;
  let A be OSSubset of OU0, B be OSSubset of OU0;
  set m = ((OSMSubSort A)# * (the Arity of S1)).o, b = (B# * (the Arity of S1)
  ).o, d = Den(o,OU0);
  assume
A1: B in OSSubSort(A);
  then B is opers_closed by Th19;
  then B is_closed_on o;
  then
A2: rng (d|b) c= (B * (the ResultSort of S1)).o;
  b /\ m = m by A1,Th26,XBOOLE_1:28;
  then d|m = (d|b)|m by RELAT_1:71;
  then rng (d|m) c= rng(d|b) by RELAT_1:70;
  hence thesis by A2;
end;
