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 Th28:
  for o be OperSymbol of S1 for A be OSSubset of OU0 holds rng ((
  Den(o,OU0))|(((OSMSubSort A)# * (the Arity of S1)).o)) c= ((OSMSubSort A) * (
  the ResultSort of S1)).o
proof
  let o be OperSymbol of S1;
  let A be OSSubset of OU0;
  let x be object;
  assume that
A1: x in rng ((Den(o,OU0))|(((OSMSubSort A)# * (the Arity of S1)).o)) and
A2: not x in ((OSMSubSort A) * (the ResultSort of S1)).o;
  set r = the_result_sort_of o;
A3: r = (the ResultSort of S1).o & dom (the ResultSort of S1) = the carrier'
  of S1 by FUNCT_2:def 1,MSUALG_1:def 2;
  then ((OSMSubSort A) * (the ResultSort of S1)).o = (OSMSubSort A).r by
FUNCT_1:13
    .= meet OSSubSort(A,r) by Def11;
  then consider X be set such that
A4: X in OSSubSort(A,r) and
A5: not x in X by A2,SETFAM_1:def 1;
  consider B be OSSubset of OU0 such that
A6: B in OSSubSort(A) and
A7: B.r = X by A4,Def10;
  rng (Den(o,OU0)|(((OSMSubSort A)# * (the Arity of S1)).o)) c= (B * (the
  ResultSort of S1)).o by A6,Th27;
  then x in (B * (the ResultSort of S1)).o by A1;
  hence contradiction by A3,A5,A7,FUNCT_1:13;
end;
