reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);
reserve t, t9, t99 for Element of dom tree_of_subformulae(F);
reserve x for set;

theorem Th23:
  t9 = t^<*m*> & (tree_of_subformulae(F)).t is universal implies (
  tree_of_subformulae(F)).t9 = the_scope_of (tree_of_subformulae(F)).t & m = 0
proof
  set G = (tree_of_subformulae(F)).t;
  set H = (tree_of_subformulae(F)).t9;
  assume that
A1: t9 = t^<*m*> and
A2: G is universal;
A3: dom <* the_scope_of G *> = Seg 1 by FINSEQ_1:def 8;
A4: succ(tree_of_subformulae(F),t) = list_of_immediate_constituents(G) & ex
  q being Element of dom tree_of_subformulae(F) st q = t & succ(
  tree_of_subformulae(F),t) = tree_of_subformulae(F)*(q succ) by Def2,
TREES_9:def 6;
   VERUM(A) is not universal by QC_LANG1:20; then
A5: G <> VERUM(A) by A2;
  G is non atomic non negative non conjunctive by A2,QC_LANG1:20;
  then list_of_immediate_constituents(G) = <* the_scope_of G *> by Def1,A5;
  then dom <* the_scope_of G *> = dom (t succ) by A4,TREES_9:37;
  then m+1 in dom <* the_scope_of G *> by A1,TREES_9:39;
  then
A6: m+1 = 0+1 by A3,FINSEQ_1:2,TARSKI:def 1;
  H is_immediate_constituent_of G by A1,Th7;
  hence thesis by A2,A6,QC_LANG2:50;
end;
