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 Th40:
  for s being SortSymbol of S
  for x being Element of X.s holds root-tree [x,s] is Element of A1,s
  proof
    let s be SortSymbol of S;
    let x be Element of X.s;
    FreeGen X is ManySortedSubset of the Sorts of A1 by Def7; then
A1: (FreeGen X).s c= (the Sorts of A1).s by PBOOLE:def 2,def 18;
    root-tree [x,s] in FreeGen(s,X) by MSAFREE:def 15; then
    root-tree [x,s] in (FreeGen X).s by MSAFREE:def 16;
    hence root-tree [x,s] is Element of A1,s by A1;
  end;
