theorem Th93:
  for S being ManySortedSign, o being set, t being DecoratedTree holds
  S variables_in ([o, the carrier of S]-tree <*t*>) = S variables_in t
proof
  let S be ManySortedSign, o be set, t be DecoratedTree;
  now
    let s be object;
    assume
A1: s in the carrier of S;
A2: t in {t} by TARSKI:def 1;
    now
      let x be object;
      rng <*t*> = {t} by FINSEQ_1:39;
      then x in (S variables_in ([o, the carrier of S]-tree <*t*>)).s iff
      ex q being DecoratedTree st q in {t} & x in (S variables_in q).s
      by A1,MSAFREE3:11;
      hence
      x in (S variables_in ([o, the carrier of S]-tree <*t*>)).s iff
      x in (S variables_in t).s by A2,TARSKI:def 1;
    end;
    hence (S variables_in ([o, the carrier of S]-tree <*t*>)).s =
    (S variables_in t).s by TARSKI:2;
  end;
  hence thesis;
end;
