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 Th41:
  for U0 be non-empty OSAlgebra of S1, U1,U2 be strict
  OSSubAlgebra of U0 holds (U1 /\ U2) "\/"_os U2 = U2
proof
  let U0 be non-empty OSAlgebra of S1, U1,U2 be strict OSSubAlgebra of U0;
  reconsider u12= the Sorts of (U1 /\ U2), u2 = the Sorts of U2 as MSSubset of
  U0 by MSUALG_2:def 9;
A1: u12 is OrderSortedSet of S1 & u2 is OrderSortedSet of S1 by OSALG_1:17;
  then reconsider u12, u2 as OSSubset of U0 by Def2;
  u12 c= the Sorts of U0 & u2 c= the Sorts of U0 by PBOOLE:def 18;
  then u12 (\/) u2 c= the Sorts of U0 by PBOOLE:16;
  then reconsider A= u12 (\/) u2 as MSSubset of U0 by PBOOLE:def 18;
  A is OrderSortedSet of S1 by A1,Th2;
  then reconsider A= u12 (\/) u2 as OSSubset of U0 by Def2;
  u12 = (the Sorts of U1) (/\) (the Sorts of U2) by MSUALG_2:def 16;
  then u12 c= u2 by PBOOLE:15;
  then
A2: A = u2 by PBOOLE:22;
  (U1 /\ U2) "\/"_os U2 = GenOSAlg(A) by Def13;
  hence thesis by A2,Th35;
end;
