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 Th14:
  t is_a_proper_prefix_of t9 implies len @((tree_of_subformulae(F)
  ).t9) < len @((tree_of_subformulae(F)).t)
proof
  set G = (tree_of_subformulae(F)).t;
  set H = (tree_of_subformulae(F)).t9;
  assume
A1: t is_a_proper_prefix_of t9;
  then
A2: t is_a_prefix_of t9;
A3: now
    consider r being FinSequence such that
A4: t9 = t^r by A2,TREES_1:1;
    reconsider r as FinSequence of NAT by A4,FINSEQ_1:36;
A5: 1 <= len r
    proof
      reconsider t1 = {} as Element of dom tree_of_subformulae(F) by TREES_1:22
;
      r <> {} & t1 is_a_prefix_of r by A1,A4,FINSEQ_1:34;
      then
A6:   t1 is_a_proper_prefix_of r;
      len t1 = 0;
      then 0 < len r by A6,TREES_1:6;
      then 0+1 <= len r by NAT_1:13;
      hence thesis;
    end;
    defpred P[set,object] means
ex t1 being Element of dom tree_of_subformulae(F)
, r1 being FinSequence, m being Nat st m = $1 & r1 = r|Seg m & t1 = t^r1 & $2 =
    (tree_of_subformulae(F)).t1;
A7: for k be Nat st k in Seg len r ex x being object st P[k,x]
    proof
      let k be Nat such that
   k in Seg len r;
      r|Seg k is FinSequence by FINSEQ_1:15;
      then consider r1 being FinSequence such that
A8:   r1 = r|Seg k;
      r1 is_a_prefix_of r by A8,TREES_1:def 1;
      then t^r1 in dom tree_of_subformulae(F) by A4,FINSEQ_6:13,TREES_1:20;
      then consider t1 being Element of dom tree_of_subformulae(F) such that
A9:  t1 = t^r1;
      ex x st x = (tree_of_subformulae(F)).t1;
      hence thesis by A8,A9;
    end;
    ex L being FinSequence st dom L = Seg len r & for k be Nat st k in
    Seg len r holds P[k,L.k] from FINSEQ_1:sch 1(A7);
    then consider L being FinSequence such that
    dom L = Seg len r and
A10: for k be Nat st k in Seg len r holds ex t1 being Element of dom
tree_of_subformulae(F), r1 being FinSequence, m being Nat st m = k & r1 = r|Seg
    m & t1 = t^r1 & L.k = (tree_of_subformulae(F)).t1;
    for k st 1 <= k & k <= len r holds ex t1 being Element of dom
tree_of_subformulae(F), r1 being FinSequence st r1 = r|Seg k & t1 = t^r1 & L.k
    = (tree_of_subformulae(F)).t1
    proof
      let k;
      assume 1 <= k & k <= len r;
      then k in Seg len r by FINSEQ_1:1;
      then ex t1 being Element of dom tree_of_subformulae(F), r1 being
      FinSequence, m being Nat st m = k & r1 = r|Seg m & t1 = t^r1 & L.k = (
      tree_of_subformulae(F)).t1 by A10;
      hence thesis;
    end;
    then consider t1 being Element of dom tree_of_subformulae(F), r1 being
    FinSequence such that
A11: r1 = r|Seg 1 and
A12: t1 = t^r1 and
    L.1 = (tree_of_subformulae(F)).t1 by A5;
    set H1 = (tree_of_subformulae(F)).t1;
A13: H1 is_immediate_constituent_of G
    proof
      ex m being Element of NAT st r1 = <*m*> by A5,A11,TREES_9:34;
      hence thesis by A12,Th7;
    end;
    r1 is_a_prefix_of r by A11,TREES_1:def 1;
    then t1 is_a_prefix_of t9 by A4,A12,FINSEQ_6:13;
    then
A14: len @ H <= len @ H1 by Th1,Th13;
    assume len @ H = len @ G;
    hence contradiction by A14,A13,QC_LANG2:51;
  end;
  len @ H <= len @ G by A2,Th1,Th13;
  hence thesis by A3,XXREAL_0:1;
end;
