reserve A,B,p,q,r for Element of LTLB_WFF,
  M for LTLModel,
  j,k,n for Element of NAT,
  i for Nat,
  X for Subset of LTLB_WFF,
  F for finite Subset of LTLB_WFF,
  f for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y,z for set,
  P,Q,R for PNPair;

theorem Th1:
  for X be set,G be non empty finite FinSequenceSet of X holds
  ex A be FinSequence st A in G &
  for B be FinSequence st B in G holds len B <= len A
  proof
    let X be set,G be non empty finite FinSequenceSet of X;
    set g = the Enumeration of G;
    deffunc F1(Nat)= len (g/.$1);
    consider f be FinSequence of NAT such that
A1: len f = len g & for n be Nat st n in dom f holds f.n = F1(n)
    from FINSEQ_2:sch 1;
    rng f c= REAL by NUMBERS:19;
    then reconsider f1 = f as FinSequence of REAL by FINSEQ_1:def 4;
    set m = max_p f1, A = g/.m;
A2: g <> {} by RLAFFIN3:def 1,RELAT_1:38;
    for B be FinSequence st B in G holds len B <= len A
    proof
      let B be FinSequence;
      set m1 = B .. g;
      m in dom f by A2,A1,RFINSEQ2:def 1;
      then A3: len A = f1.m by A1;
      assume B in G;
      then A4: B in rng g by RLAFFIN3:def 1;
      then A5: m1 in dom g by FINSEQ_4:20;
      then A6: m1 in dom f by A1,FINSEQ_3:29;
      g/.m1 = g.m1 by A5,PARTFUN1:def 6
      .= B by FINSEQ_4:19,A4;
      then len B = f1.m1 by A6,A1;
      hence len B <= len A by A6,RFINSEQ2:def 1, A2,A1,A3;
    end;
    hence thesis;
  end;
