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 Th22:
  t9 = t^<*m*> & (tree_of_subformulae(F)).t is conjunctive implies
(tree_of_subformulae(F)).t9 = the_left_argument_of (tree_of_subformulae(F)).t &
  m = 0 or (tree_of_subformulae(F)).t9 = the_right_argument_of (
  tree_of_subformulae(F)).t & m = 1
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 conjunctive;
A3: list_of_immediate_constituents(G) = <* the_left_argument_of G,
  the_right_argument_of G *> by A2,Def1;
  len <* the_left_argument_of G, the_right_argument_of G *> = 2 by FINSEQ_1:44;
  then
A4: dom <* the_left_argument_of G, the_right_argument_of G *> = Seg 2 by
FINSEQ_1:def 3;
A5: succ(tree_of_subformulae(F),t) = list_of_immediate_constituents(G) by Def2;
  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 TREES_9:def 6;
  then
  dom <* the_left_argument_of G, the_right_argument_of G *> = dom (t succ)
  by A5,A3,TREES_9:37;
  then
  m+1 in dom <* the_left_argument_of G, the_right_argument_of G *> by A1,
TREES_9:39;
  then
A6: m+1 = 0+1 or m+1 = 1+1 by A4,FINSEQ_1:2,TARSKI:def 2;
  per cases by A6;
  suppose
A7: m = 0;
    H = succ(tree_of_subformulae(F),t).(m+1) by A1,TREES_9:40
      .= the_left_argument_of G by A5,A3,A7;
    hence thesis by A7;
  end;
  suppose
A8: m = 1;
    H = succ(tree_of_subformulae(F),t).(m+1) by A1,TREES_9:40
      .= the_right_argument_of G by A5,A3,A8;
    hence thesis by A8;
  end;
end;
