
theorem Th10:
  for G being binary with_terminals with_nonterminals non empty
  DTConstrStr, ts being FinSequence of TS G, nt being Symbol of G st nt ==>
  roots ts holds nt is NonTerminal of G & dom ts = {1, 2} & 1 in dom ts & 2 in
  dom ts & ex tl, tr being Element of TS G st roots ts = <*root-label tl,
root-label tr*> & tl = ts.1 & tr = ts.2 & nt-tree ts = nt-tree (tl, tr) & tl in
  rng ts & tr in rng ts
proof
  let G be binary with_terminals with_nonterminals non empty DTConstrStr, ts
  be FinSequence of TS G, nt be Symbol of G;
  assume
A1: nt ==> roots ts;
  then consider rtl, rtr being Symbol of G such that
A2: roots ts = <* rtl, rtr *> by Def4;
  nt in {s where s is Symbol of G:ex rts being FinSequence st s ==> rts} by A1;
  hence nt is NonTerminal of G by LANG1:def 3;
A3: len <*rtl, rtr*> = 2 by FINSEQ_1:44;
A4: dom <*rtl, rtr*> = dom ts by A2,TREES_3:def 18;
  hence dom ts = {1, 2} by A3,FINSEQ_1:2,def 3;
  hence
A5: 1 in dom ts & 2 in dom ts by TARSKI:def 2;
  then consider tl being DecoratedTree such that
A6: tl = ts.1 and
A7: <*rtl, rtr*>.1 = tl.{} by A2,TREES_3:def 18;
A8: rng ts c= TS G & tl in rng ts by A5,A6,FINSEQ_1:def 4,FUNCT_1:def 3;
  consider tr being DecoratedTree such that
A9: tr = ts.2 and
A10: <*rtl, rtr*>.2 = tr.{} by A2,A5,TREES_3:def 18;
  tr in rng ts by A5,A9,FUNCT_1:def 3;
  then reconsider tl, tr as Element of TS(G) by A8;
  take tl, tr;
  thus roots ts = <*root-label tl, root-label tr*> by A2,A7,A10;
  Seg len <*rtl, rtr*> = dom <*rtl, rtr*> by FINSEQ_1:def 3
    .= Seg len ts by A4,FINSEQ_1:def 3;
  then len ts = 2 by A3,FINSEQ_1:6;
  then ts = <*tl, tr*> by A6,A9,FINSEQ_1:44;
  hence tl = ts.1 & tr = ts.2 & nt-tree ts = nt-tree (tl, tr) by
TREES_4:def 6;
  thus thesis by A5,A6,A9,FUNCT_1:def 3;
end;
