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;
reserve x,y for set;

theorem Th25:
  (tree_of_subformulae(F)).t is conjunctive implies t^<*0*> in dom
  tree_of_subformulae(F) & (tree_of_subformulae(F)).(t^<*0*>) =
  the_left_argument_of (tree_of_subformulae(F)).t & t^<*1*> in dom
  tree_of_subformulae(F) & (tree_of_subformulae(F)).(t^<*1*>) =
  the_right_argument_of (tree_of_subformulae(F)).t
proof
  set G = (tree_of_subformulae(F)).t;
  assume
A1: G is conjunctive;
  (tree_of_subformulae(F))*(t succ) = (succ(tree_of_subformulae(F),t))
  proof
    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;
    hence thesis;
  end;
  then
A2: dom (t succ) = dom (succ(tree_of_subformulae(F),t)) by TREES_9:37
    .= dom (list_of_immediate_constituents(G)) by Def2
    .= dom <* the_left_argument_of G, the_right_argument_of G *> by A1,Def1
    .= {1,2} by Lm1,FINSEQ_1:2;
A3: 0+1 in {1,2} by TARSKI:def 2;
  then t^<*0*> in dom tree_of_subformulae(F) by A2,TREES_9:39;
  then
  (tree_of_subformulae(F)).(t^<*0*>) = (succ(tree_of_subformulae(F),t)).(0
  +1) by TREES_9:40
    .= (list_of_immediate_constituents(G)).1 by Def2
    .= <* the_left_argument_of G, the_right_argument_of G *>.1 by A1,Def1
    .= the_left_argument_of G;
  hence
  t^<*0*> in dom tree_of_subformulae(F) & (tree_of_subformulae(F)).(t^<*0
  *>) = the_left_argument_of (tree_of_subformulae(F)).t by A2,A3,TREES_9:39;
A4: 1+1 in {1,2} by TARSKI:def 2;
  then t^<*1*> in dom tree_of_subformulae(F) by A2,TREES_9:39;
  then
  (tree_of_subformulae(F)).(t^<*1*>) = (succ(tree_of_subformulae(F),t)).(
  1+1) by TREES_9:40
    .= (list_of_immediate_constituents(G)).2 by Def2
    .= <* the_left_argument_of G, the_right_argument_of G *>.2 by A1,Def1
    .= the_right_argument_of G;
  hence thesis by A2,A4,TREES_9:39;
end;
