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 Th61:
  H is_immediate_constituent_of F implies H is_proper_subformula_of F
proof
  assume
A1: H is_immediate_constituent_of F;
  thus H is_subformula_of F
  proof
    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;
  assume H = F;
  then len H = len F;
  hence contradiction by A1,Th60;
end;
