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 Th38:
  for U0 be non-empty OSAlgebra of S1, U1 be OSSubAlgebra of U0, B
  be OSSubset of U0 st B = the Sorts of U0 holds GenOSAlg(B) "\/"_os U1 =
  GenOSAlg(B)
proof
  let U0 be non-empty OSAlgebra of S1, U1 be OSSubAlgebra of U0, B be OSSubset
  of U0;
A1: the Sorts of U1 is MSSubset of U0 by MSUALG_2:def 9;
  assume B = the Sorts of U0;
  then the Sorts of U1 c= B by A1,PBOOLE:def 18;
  then B (\/) the Sorts of U1 = B by PBOOLE:22;
  hence thesis by Th37;
end;
