reserve Al for QC-alphabet;
reserve a,b,b1 for object,
  i,j,k,n for Nat,
  p,q,r,s for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  P for QC-pred_symbol of k,Al,
  l,ll for CQC-variable_list of k,Al,
  Sub,Sub1 for CQC_Substitution of Al,
  S,S1,S2 for Element of CQC-Sub-WFF(Al),
  P1,P2 for Element of QC-pred_symbols(Al);
reserve F1,F2,F3 for QC-formula of Al,
  L for FinSequence;

theorem
  for L being PATH of F1,F2 st F1 is_subformula_of F2 & 1 <= i & i <=
  len L holds ex F3 st F3 = L.i & F3 is_subformula_of F2
proof
  let L be PATH of F1,F2;
  set n = len L;
  assume that
A1: F1 is_subformula_of F2 and
A2: 1 <= i and
A3: i <= n;
  n+1 <= n+i by A2,XREAL_1:6;
  then n+1+(-1) <= n+i+(-1) by XREAL_1:6;
  then
A4: n+(-i) <= n-1+i+(-i) by XREAL_1:6;
  i+(-i) <= n+(-i) by A3,XREAL_1:6;
  then reconsider l = n-i as Element of NAT by INT_1:3;
  defpred P[Nat] means $1 <= n-1 implies ex F3 st F3 = L.(n-$1) &
  F3 is_subformula_of F2;
A5: for k st P[k] holds P[k+1]
  proof
    let k such that
A6: P[k];
    assume
A7: k+1 <= n-1;
    then k+1+1 <= n-1+1 by XREAL_1:6;
    then
A8: 2+k+(-k) <= n+(-k) by XREAL_1:6;
    then reconsider j = n-k as Element of NAT by INT_1:3;
    n <= n+k by NAT_1:11;
    then n+(-k) <= n+k+(-k) by XREAL_1:6;
    then
A9: j-1 < n by XREAL_1:146,XXREAL_0:2;
A10: 1+1+(-1) <= j+(-1) by A8,XREAL_1:6;
    then reconsider j1 = j-1 as Element of NAT by INT_1:3;
    j1+1 = j;
    then
A11: ex G1,H1 being Element of QC-WFF(Al) st L.j1 = G1 & L.j = H1 & G1
    is_immediate_constituent_of H1 by A1,A10,A9,Def5;
    then reconsider F3 = L.j1 as QC-formula of Al;
    take F3;
    k < k+1 by NAT_1:13;
    then F3 is_proper_subformula_of F2 by A6,A7,A11,QC_LANG2:63,XXREAL_0:2;
    hence thesis by QC_LANG2:def 21;
  end;
  L.n = F2 by A1,Def5;
  then
A12: P[0];
  for k holds P[k] from NAT_1:sch 2(A12,A5);
  then ex F3 st F3 = L.(n-l) & F3 is_subformula_of F2 by A4;
  hence thesis;
end;
