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 Th39:
  for U0 being non-empty OSAlgebra of S1, U1,U2 be OSSubAlgebra of
  U0 holds U1 "\/"_os U2 = U2 "\/"_os U1
proof
  let U0 be non-empty OSAlgebra of S1, U1,U2 be OSSubAlgebra of U0;
  reconsider u1= the Sorts of U1, u2= the Sorts of U2 as MSSubset of U0 by
MSUALG_2:def 9;
  u1 c= the Sorts of U0 & u2 c= the Sorts of U0 by PBOOLE:def 18;
  then u1 (\/) u2 c= the Sorts of U0 by PBOOLE:16;
  then reconsider A1 = u1 (\/) u2 as MSSubset of U0 by PBOOLE:def 18;
  u1 is OrderSortedSet of S1 & u2 is OrderSortedSet of S1 by OSALG_1:17;
  then A1 is OrderSortedSet of S1 by Th2;
  then reconsider A = A1 as OSSubset of U0 by Def2;
  U1 "\/"_os U2 = GenOSAlg(A) by Def13;
  hence thesis by Def13;
end;
