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 Th32:
  for P be consistent complete PNPair,T be pnptree of P st
  A 'U' B in rng P`1 ex R be Element of rngr T st B in rng R`1
  proof
    let P be consistent complete PNPair,T be pnptree of P;
    set nb = 'not' B,gnb = 'G' nb,xgnb = 'X' gnb,aub = A 'U' B;
    set f = the Enumeration of (rngr T)^;
    set xaf = 'X' alt(f);
A1: rng f = (rngr T)^ by RLAFFIN3:def 1;
    {} in dom T by TREES_1:22;then
    T.{} in rng T by FUNCT_1:3;then
A2: P in rng T by Def11;
    reconsider f as FinSequence of l;
    assume
A3: aub in rng P`1;
    assume
A4: for R be Element of rngr T holds not B in rng R`1;
    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
A5:   f/.i = R^ and
A6:   R in rngr T;
      B in rng R`2 by Th31,A3,A4,A6;
      hence {}l |- (f/.i) => nb by LTLAXIO3:29,A5;
    end;then
A7: {}l |- alt(f) => nb by LTLAXIO2:57;
    ('X' (alt(f) => gnb)) => (xaf => xgnb) in LTL_axioms by LTLAXIO1:def 17;
    then A8: {}l |- ('X' (alt(f) => gnb)) => (xaf => xgnb) by LTLAXIO1:42;
    {}l |- alt(f) => xaf by Th29,A1;then
    {}l |- 'X' (alt(f) => gnb) by LTLAXIO1:45,A7,LTLAXIO1:44;then
A9: {}l |- xaf => xgnb by A8,LTLAXIO1:43;
    consider R be object such that
A10: R in untn P by XBOOLE_0:def 1;
    reconsider R as Element of untn P by A10;
    set xr = 'X' (R^);
    set g = the Enumeration of (comp R)^;
    reconsider g as FinSequence of l;
A11: rng g = (comp R)^ by RLAFFIN3:def 1;
    (rngr T)^ = (comp R \/ rngr T)^ by Th28,XBOOLE_1:12,A2
    .= (comp R)^ \/ (rngr T)^ by Th10;
    then  alt(g) => alt(f) is ctaut by XBOOLE_1:7,A11,A1 ,LTLAXIO2:30;
    then  alt(g) => alt(f) in LTL_axioms by LTLAXIO1:def 17;
    then A12: {}l |- alt(g) => alt(f) by LTLAXIO1:42;
A13: {}l |- P^ => xr by Th18;
     ('X' (R^ => alt(f))) => (xr => xaf) in LTL_axioms by LTLAXIO1:def 17;
     then A14: {}l |- ('X' (R^ => alt(f))) => (xr => xaf) by LTLAXIO1:42;
     {}l |- R^ => alt(g) by A11,Th17;
     then  {}l |- 'X' (R^ => alt(f)) by A12,LTLAXIO1:47,LTLAXIO1:44;
     then  {}l |- xr => xaf by LTLAXIO1:43,A14;then
     {}l |- P^ => xaf by LTLAXIO1:47,A13;then
A15: {}l |- P^ => xgnb by LTLAXIO1:47,A9;
A16:  {}l |- P^ => aub by LTLAXIO3:28,A3;
     aub => ('X' ('F' B)) in LTL_axioms by LTLAXIO1:def 17;then
A17: {}l |- aub => ('X' ('F' B)) by LTLAXIO1:42;
     ('X' ('F' B)) => ('not' xgnb) in LTL_axioms by LTLAXIO1:def 17;
     then  {}l |- ('X' ('F' B)) => ('not' xgnb) by LTLAXIO1:42;
     then  {}l |- aub => ('not' xgnb) by A17,LTLAXIO1:47;
     then  {}l |- P^ => ('not' xgnb) by LTLAXIO1:47,A16;
     then  {}l |- P^ => (xgnb '&&' ('not' xgnb)) by A15,LTLAXIO2:52;
     then  {}l |- 'not' (P^) by LTLAXIO2:55;
     hence contradiction by LTLAXIO3:def 10;
   end;
