reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;

theorem Th29:
  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,Th28;
end;
