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;
reserve t for Element of dom tree_of_subformulae(F),
  s for Element of dom tree_of_subformulae(G);
reserve t for Element of dom tree_of_subformulae(F),
  s for FinSequence;

theorem Th28:
  t in F-entry_points_in_subformula_tree_of G & t^s in F
  -entry_points_in_subformula_tree_of H implies s in G
  -entry_points_in_subformula_tree_of H
proof
  defpred P[Nat] means for F,G,H,t,s holds G = (tree_of_subformulae
  (F)).t & t^s in F-entry_points_in_subformula_tree_of H & len s = $1 implies s
  in G-entry_points_in_subformula_tree_of H;
A1: for k st P[k] holds P[k + 1]
  proof
    let k such that
A2: P[k];
    thus P[k + 1]
    proof
      let F,G,H,t,s;
      assume that
A3:   G = (tree_of_subformulae(F)).t and
A4:   t^s in F-entry_points_in_subformula_tree_of H and
A5:   len s = k+1;
      consider v being FinSequence, x being set such that
A6:   s = v^<*x*> and
A7:   len v = k by A5,TREES_2:3;
      F-entry_points_in_subformula_tree_of H = { t1 where t1 is Element
of dom tree_of_subformulae(F) : (tree_of_subformulae(F)).t1 = H } by Th19;
      then consider t99 such that
A8:   t99 = t^s and
A9:   (tree_of_subformulae(F)).t99 = H by A4;
      reconsider s1 = s as FinSequence of NAT by A8,FINSEQ_1:36;
A10:  s1 = v^<*x*> by A6;
      then reconsider u = <*x*> as FinSequence of NAT by FINSEQ_1:36;
      reconsider v as FinSequence of NAT by A10,FINSEQ_1:36;
A11:  rng u c= NAT by FINSEQ_1:def 4;
A12:  1 in {1} by TARSKI:def 1;
      dom u = Seg 1 & u.1 = x by FINSEQ_1:def 8;
      then x in rng u by A12,FINSEQ_1:2,FUNCT_1:def 3;
      then reconsider m = x as Element of NAT by A11;
      consider t9 being FinSequence of NAT such that
A13:  t9 = t^v;
A14:  t99 = t9^<*m*> by A6,A8,A13,FINSEQ_1:32;
      then t9 is_a_prefix_of t99 by TREES_1:1;
      then reconsider t9 as Element of dom tree_of_subformulae(F) by TREES_1:20
;
      consider H9 such that
A15:  H9 = (tree_of_subformulae(F)).t9;
      t^v in F-entry_points_in_subformula_tree_of H9 by A13,A15,Def3;
      then
A16:  v in G-entry_points_in_subformula_tree_of H9 by A2,A3,A7;
      G-entry_points_in_subformula_tree_of H9 = { v1 where v1 is Element
of dom tree_of_subformulae(G) : (tree_of_subformulae(G)).v1 = H9 } by Th19;
      then
A17:  ex v1 being Element of dom tree_of_subformulae(G) st v = v1 & (
      tree_of_subformulae(G)).v1 = H9 by A16;
      then reconsider v as Element of dom tree_of_subformulae(G);
A18:  H is_immediate_constituent_of H9 by A9,A14,A15,Th7;
      H = (tree_of_subformulae(G)).(v^<*m*>) & v^<*m*> in dom
      tree_of_subformulae(G)
      proof
A19:    H9 <> VERUM(A) by A18,QC_LANG2:41;
        now
          per cases by A18,A19,QC_LANG1:9,QC_LANG2:47;
          suppose
A20:        H9 is negative;
            then H = the_argument_of H9 & m = 0 by A9,A14,A15,Th21;
            hence thesis by A17,A20,Th24;
          end;
          suppose
A21:        H9 is conjunctive;
            then H = the_left_argument_of H9 & m = 0 or H =
            the_right_argument_of H9 & m = 1 by A9,A14,A15,Th22;
            hence thesis by A17,A21,Th25;
          end;
          suppose
A22:        H9 is universal;
            then H = the_scope_of H9 & m = 0 by A9,A14,A15,Th23;
            hence thesis by A17,A22,Th26;
          end;
        end;
        hence thesis;
      end;
      hence thesis by A6,Def3;
    end;
  end;
A23: P[0]
  proof
    let F,G,H,t,s;
    assume that
A24: G = (tree_of_subformulae(F)).t and
A25: t^s in F-entry_points_in_subformula_tree_of H and
A26: len s = 0;
A27: s = {} by A26;
    then reconsider s9 = s as Element of dom tree_of_subformulae(G) by
TREES_1:22;
A28: G = (tree_of_subformulae(G)).s9 by A27,Def2;
    F-entry_points_in_subformula_tree_of H = { t1 where t1 is Element of
    dom tree_of_subformulae(F) : (tree_of_subformulae(F)).t1 = H } by Th19;
    then ex t9 st t9 = t^s & (tree_of_subformulae(F)).t9 = H by A25;
    then H = G by A24,A27,FINSEQ_1:34;
    hence thesis by A28,Def3;
  end;
  for k holds P[k] from NAT_1:sch 2(A23,A1);
  then
A29: G = (tree_of_subformulae(F)).t & t^s in F
  -entry_points_in_subformula_tree_of H & len s = len s implies s in G
  -entry_points_in_subformula_tree_of H;
  assume t in F-entry_points_in_subformula_tree_of G & t^s in F
  -entry_points_in_subformula_tree_of H;
  hence thesis by A29,Def3;
end;
