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 Th55:
  H is_proper_subformula_of F implies ex G st G is_immediate_constituent_of 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;
  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
A5: n = 2 + k by NAT_1:10;
  reconsider k as Nat;
  1 + 1 + k = (1 + k) + 1;
  then 1 + k < n by A5,NAT_1:13;
  then consider H1,F1 be Element of QC-WFF(A) such that
  L.(1 + k) = H1 and
A6: L.(1 + k + 1) = F1 & H1 is_immediate_constituent_of F1 by A4,NAT_1:11;
  take H1;
  thus thesis by A3,A5,A6;
end;
