reserve A for QC-alphabet;
reserve sq for FinSequence,
  x,y,z for bound_QC-variable of A,
  p,q,p1,p2,q1 for Element of QC-WFF(A);
reserve s,t for bound_QC-variable of A;
reserve F,G,H,H1 for Element of QC-WFF(A);
reserve x,y,z for bound_QC-variable of A,
  k,n,m for Nat,
  P for ( QC-pred_symbol of k, A),
  V for QC-variable_list of k, A;
reserve L,L9 for FinSequence;

theorem Th54:
  H is_proper_subformula_of F implies len @H < len @F
proof
  given n,L such that
A1: 1 <= n and
  len L = n and
A2: L.1 = H and
A3: L.n = F and
A4: for k st 1 <= k & k < n ex H1,F1 being Element of QC-WFF(A) st L.k = H1
  & L.(k + 1) = F1 & H1 is_immediate_constituent_of F1;
  defpred P[Nat] means 1 <= $1 & $1 < n implies for H1 st L.($1 + 1
  ) = H1 holds len @H < len @H1;
A5: for k st P[k] holds P[k+1]
  proof
    let k such that
A6: 1 <= k & k < n implies for H1 st L.(k + 1) = H1 holds len @H < len @H1 and
A7: 1 <= k + 1 and
A8: k + 1 < n;
    consider F1,G be Element of QC-WFF(A) such that
A9: L.(k + 1) = F1 and
A10: L.(k + 1 + 1) = G & F1 is_immediate_constituent_of G by A4,A7,A8;
    let H1 such that
A11: L.(k + 1 + 1) = H1;
A12: now
      given m be Nat such that
A13:  k = m + 1;
      len @H < len @F1 by A6,A8,A9,A13,NAT_1:11,13;
      hence thesis by A11,A10,Th51,XXREAL_0:2;
    end;
    k = 0 implies len @H < len @H1 by A2,A11,A9,A10,Th51;
    hence thesis by A12,NAT_1:6;
  end;
  assume H <> F;
  then 1 < n by A1,A2,A3,XXREAL_0:1;
  then 1 + 1 <= n by NAT_1:13;
  then consider k being Nat such that
A14: n = 2 + k by NAT_1:10;
A15: P[0];
A16: for k holds P[k] from NAT_1:sch 2(A15,A5);
  reconsider k as Nat;
A17: 1 + 1 + k = (1 + k) + 1;
  then 1 + k < n by A14,NAT_1:13;
  hence thesis by A3,A16,A14,A17,NAT_1:11;
end;
