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 Th42:
  for t being Element of A0 holds t is Term of S,X
  proof
    let t be Element of A0;
    consider s being object such that
A1: s in dom the Sorts of A0 & t in (the Sorts of A0).s by CARD_5:2;
    reconsider s as SortSymbol of S by A1;
    the Sorts of A0 is ManySortedSubset of the Sorts of Free(S,X)
    by Def6; then
    (the Sorts of A0).s c= (the Sorts of Free(S,X)).s by PBOOLE:def 2,def 18;
    then t is Element of (the Sorts of Free(S,X)).s by A1; then
    t is Element of FreeMSA X by MSAFREE3:31;
    hence t is Term of S,X by MSAFREE3:6;
  end;
