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;
reserve T for pnptree of P,t for Node of T;

theorem Th29:
  for P be consistent complete PNPair,T be pnptree of P,
  f be FinSequence of LTLB_WFF st rng f = (rngr T)^ holds
  {}LTLB_WFF |- ('not' ((con nega f)/.(len con nega f))) =>
  ('X' ('not' ((con nega f)/.(len con nega f))))
  proof
   let P be consistent complete PNPair,T be pnptree of P,f be FinSequence of l;
   assume
A1: rng f = (rngr T)^;
A2: now
      let R be Element of rng T;
      consider Q be object such that
A3:   Q in untn R by XBOOLE_0:def 1;
      reconsider Q as Element of untn R by A3;
      consider g be FinSequence such that
A4:   rng g = (comp Q)^ and
      g is one-to-one by FINSEQ_4:58;
      reconsider g as FinSequence of l by A4,FINSEQ_1:def 4;
A5:   {}l |- R^ => 'X' (Q^) by Th18;
      reconsider Q as consistent PNPair;
      now
        let j be Nat;
        assume j in dom g;
        then g/.j in (comp Q)^ by PARTFUN2:2,A4;
        then consider S being PNPair such that
A6:     g/.j = S^ and
A7:     S in comp Q;
        comp Q c= rngr T by Th28;
        then S^ in (rngr T)^ by A7;
        then consider k such that
A8:     k in dom f and
A9:     f/.k = g/.j by A6,A1,PARTFUN2:2;
        f/.k => alt(f) is ctaut by LTLAXIO2:29,A8;
        then f/.k => alt(f) in LTL_axioms by LTLAXIO1:def 17;
        hence {}l |- (g/.j) => alt(f) by LTLAXIO1:42,A9;
      end;
      then A10: {}l |- alt(g) => alt(f) by LTLAXIO2:57;
      ('X' (Q^ => alt(f))) => (('X' (Q^)) => ('X' alt(f))) in LTL_axioms
      by LTLAXIO1:def 17;then
A11:  {}l |- ('X' (Q^ => alt(f))) => (('X' (Q^)) => ('X' alt(f)))
      by LTLAXIO1:42;
      {}l |- Q^ => alt(g) by Th17,A4;
      then {}l |- 'X' (Q^ => alt(f)) by A10,LTLAXIO1:47,LTLAXIO1:44;
      then {}l |- ('X' (Q^)) => ('X' alt(f)) by A11,LTLAXIO1:43;
      hence {}l |- R^ => ('X' alt(f)) by LTLAXIO1:47,A5;
    end;
    now
      let i be Nat;
      assume i in dom f;
      then f/.i in (rngr T)^ by PARTFUN2:2,A1;
      then consider R such that
A12:  f/.i = R^ and
A13:  R in rngr T;
      ex t be Node of T st R = T.t & t <> {} by A13;
      then R in rng T by FUNCT_1:3;
      hence {}l |- (f/.i) => ('X' alt(f)) by A2,A12;
    end;
    hence {}l |- alt(f) => ('X' alt(f)) by LTLAXIO2:57;
  end;
