reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;

theorem Th63:
  H is_proper_subformula_of F implies ex G st G is_immediate_constituent_of F
proof
  assume H is_subformula_of F;
  then consider 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 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 be Nat such that
A5: n = 2 + k by NAT_1:10;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  1 + 1 + k = (1 + k) + 1;
  then 1 + k < n by A5,NAT_1:13;
  then consider H1,F1 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;
