reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;
reserve
  A0 for (X,S)-terms non-empty MSAlgebra over S,
  A1 for all_vars_including (X,S)-terms MSAlgebra over S,
  A2 for all_vars_including inheriting_operations (X,S)-terms MSAlgebra over S,
  A for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S;

theorem Th39:
  (for t being Element of A0 holds t is Element of Free(S,X)) &
  for s being SortSymbol of S
  for t being Element of A0,s holds t is Element of Free(S,X),s
  proof
A1: the Sorts of A0 is MSSubset of Free(S,X) by Def6;
    then Union the Sorts of A0 c= Union the Sorts of Free(S,X)
    by Th1,PBOOLE:def 18;
    hence for t being Element of A0 holds t is Element of Free(S,X);
    let s be SortSymbol of S;
    let t be Element of A0,s;
    t in (the Sorts of A0).s &
    (the Sorts of A0).s c= (the Sorts of Free(S,X)).s
    by A1,PBOOLE:def 2,def 18;
    hence thesis;
  end;
