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 Th18:
  for A being OSSubset of OU0 holds OSSubSort(A) c= OSSubSort(OU0)
proof
  let A be OSSubset of OU0;
  let x be object;
  assume x in OSSubSort(A);
  then consider x1 being Element of SubSort(A) such that
A1: x1 = x and
A2: x1 is OrderSortedSet of S1;
  x1 in SubSort(A) & SubSort(A) c= SubSort(OU0) by MSUALG_2:39;
  then reconsider x2 = x1 as Element of SubSort(OU0);
  x2 in { y where y is Element of SubSort(OU0): y is OrderSortedSet of S1
  } by A2;
  hence thesis by A1;
end;
