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;
