theorem
  for S being ManySortedSign, o being set holds
  S variables_in ([o, the carrier of S]-tree {}) = EmptyMS the carrier of S
proof
  let S be ManySortedSign, o be set;
  now
    let s be object;
    assume
A1: s in the carrier of S;
    now
      let x be object;
      rng {} = {};
      then x in (S variables_in ([o, the carrier of S]-tree {})).s iff
      ex q being DecoratedTree st q in {} & x in (S variables_in q).s
      by A1,MSAFREE3:11;
      hence x in (S variables_in ([o, the carrier of S]-tree {})).s iff
      x in (EmptyMS the carrier of S).s;
    end;
    hence (S variables_in ([o, the carrier of S]-tree {})).s =
    (EmptyMS the carrier of S).s by TARSKI:2;
  end;
  hence thesis;
end;
