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 Th52:
  H is_immediate_constituent_of F implies H is_subformula_of F
proof
  assume
A1: H is_immediate_constituent_of F;
  take n = 2 , L = <* H,F *>;
  thus 1 <= n;
  thus len L = n by FINSEQ_1:44;
  thus L.1 = H & L.n = F;
  let k;
  assume that
A2: 1 <= k and
A3: k < n;
  take H,F;
  k < 1 + 1 by A3;
  then k <= 1 by NAT_1:13;
  then k = 1 by A2,XXREAL_0:1;
  hence L.k = H & L.(k + 1) = F;
  thus thesis by A1;
end;
