reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th14:
  for U1 being monotone OSAlgebra of S1, U2 being OSSubAlgebra of
  U1 holds U2 is monotone
proof
  let U1 be monotone OSAlgebra of S1, U2 be OSSubAlgebra of U1;
  let o1,o2 be OperSymbol of S1 such that
A1: o1 <= o2;
A2: Args(o1,U2) c= Args(o2,U2) by A1,OSALG_1:26;
  the Sorts of U2 is MSSubset of U1 & the Sorts of U2 is OrderSortedSet of
  S1 by MSUALG_2:def 9,OSALG_1:17;
  then reconsider B = the Sorts of U2 as OSSubset of U1 by OSALG_2:def 2;
A3: B is opers_closed by MSUALG_2:def 9;
  then
A4: B is_closed_on o1 by MSUALG_2:def 6;
A5: B is_closed_on o2 by A3,MSUALG_2:def 6;
A6: Den(o2,U2) = (the Charact of U2).o2 by MSUALG_1:def 6
    .= Opers(U1,B).o2 by MSUALG_2:def 9
    .= o2/.B by MSUALG_2:def 8
    .= (Den(o2,U1)) | ((B# * the Arity of S1).o2) by A5,MSUALG_2:def 7
    .= (Den(o2,U1)) | Args(o2,U2) by MSUALG_1:def 4;
A7: Den(o1,U2) = (the Charact of U2).o1 by MSUALG_1:def 6
    .= Opers(U1,B).o1 by MSUALG_2:def 9
    .= o1/.B by MSUALG_2:def 8
    .= (Den(o1,U1)) | ((B# * the Arity of S1).o1) by A4,MSUALG_2:def 7
    .= (Den(o1,U1)) | Args(o1,U2) by MSUALG_1:def 4;
  Den(o2,U1)|Args(o1,U1) = Den(o1,U1) by A1,OSALG_1:def 21;
  then Den(o1,U2) = Den(o2,U1)| ( Args(o1,U1) /\ Args(o1,U2)) by A7,RELAT_1:71
    .= Den(o2,U1) | Args(o1,U2) by MSAFREE3:37,XBOOLE_1:28
    .= Den(o2,U1) | ( Args(o2,U2) /\ Args(o1,U2) ) by A2,XBOOLE_1:28
    .= Den(o2,U2) | Args(o1,U2) by A6,RELAT_1:71;
  hence thesis;
end;
