reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem Th19:
  for A be MSSubset of U0 holds rng ((Den(o,U0))|(((MSSubSort A)#
  * (the Arity of S)).o)) c= ((MSSubSort A) * (the ResultSort of S)).o
proof
  let A be MSSubset of U0;
  let x be object;
  assume that
A1: x in rng ((Den(o,U0))|(((MSSubSort A)# * (the Arity of S)).o)) and
A2: not x in ((MSSubSort A) * (the ResultSort of S)).o;
  set r = the_result_sort_of o;
A3: r = (the ResultSort of S).o & dom (the ResultSort of S) = the carrier'
  of S by FUNCT_2:def 1,MSUALG_1:def 2;
  then ((MSSubSort A) * (the ResultSort of S)).o = (MSSubSort A).r by
FUNCT_1:13
    .= meet SubSort(A,r) by Def14;
  then consider X be set such that
A4: X in SubSort(A,r) and
A5: not x in X by A2,SETFAM_1:def 1;
  consider B be MSSubset of U0 such that
A6: B in SubSort(A) and
A7: B.r = X by A4,Def13;
  rng (Den(o,U0)|(((MSSubSort A)# * (the Arity of S)).o)) c= (B * (the
  ResultSort of S)).o by A6,Th18;
  then x in (B * (the ResultSort of S)).o by A1;
  hence contradiction by A3,A5,A7,FUNCT_1:13;
end;
