
theorem Th5:
  for s being State of SCM, nt being NonTerminal of SCM-AE, tl, tr
  being bin-term holds (nt-tree(tl, tr))@s = nt-Meaning_on(tl@s, tr@s)
proof
  let s be State of SCM, nt be NonTerminal of SCM-AE, tl, tr be bin-term;
  consider f being Function of TS SCM-AE, INT such that
A1: (nt-tree(tl, tr))@s = f.(nt-tree(tl,tr)) and
A2: for t being Terminal of SCM-AE holds f.(root-tree t) = s.t and
A3: for nt being NonTerminal of SCM-AE, tl, tr being bin-term, rtl, rtr
being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*
rtl, rtr *> for xl, xr being Element of INT st xl = f.tl & xr = f.tr holds f.(
  nt-tree (tl, tr)) = nt-Meaning_on (xl, xr) by Def9;
A4: nt ==> <* root-label tl, root-label tr *> by Def1,Lm3;
  tl@s = f.tl & tr@s = f.tr by A2,A3,Def9;
  hence thesis by A1,A3,A4;
end;
