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 Th24: for P be consistent complete PNPair,Q be Element of compn P st
A 'U' B in rng P`2 holds B in rng Q`2 & (A in rng Q`2 or A 'U' B in rng Q`2)
proof
  let P be consistent complete PNPair,Q be Element of compn P;
  set aub = A 'U' B,nb = 'not' B, na = 'not' A,au = A '&&' aub;
  ('not' untn(A,B)) => (nb '&&' 'not' au) is ctaut by LTLAXIO2:36;then
  ('not' untn(A,B)) => (nb '&&' 'not' au) in LTL_axioms by LTLAXIO1:def 17;
  then A1: {}l |- ('not' untn(A,B)) => (nb '&&' 'not' au) by LTLAXIO1:42;
  assume aub in rng P`2;
  then A2: untn(A,B) in rng Q`2 by Th21;
  then A3: untn(A,B) in rng Q by XBOOLE_0:def 3;
  then A4: A in rng Q by Th8;
  {}l |- Q^ => 'not' untn(A,B) by LTLAXIO3:29,A2;
  then A5: {}l |- Q^ => (nb '&&' 'not' au) by A1,LTLAXIO1:47;
  (nb '&&' 'not' au) => nb is ctaut by LTLAXIO2:27;
  then (nb '&&' 'not' au) => nb in LTL_axioms by LTLAXIO1:def 17;
  then {}l |- (nb '&&' 'not' au) => nb by LTLAXIO1:42;
  then A6: {}l |- Q^ => nb by LTLAXIO1:47,A5;
  (nb '&&' 'not' au) => ('not' au) is ctaut by LTLAXIO2:28;then
  (nb '&&' 'not' au) => ('not' au) in LTL_axioms by LTLAXIO1:def 17;
  then {}l |- (nb '&&' 'not' au) => ('not' au) by LTLAXIO1:42;
  then A7: {}l |- Q^ => ('not' au) by LTLAXIO1:47,A5;
A8: B in rng Q by A3,Th8;
A9: aub in rng Q by A3,Th8;
    assume
A10: not B in rng Q`2 or not (A in rng Q`2 or aub in rng Q`2);
     per cases by A10;
     suppose not B in rng Q`2;
       then B in rng Q`1 by A8,XBOOLE_0:def 3;
       then {}l |- Q^ => B by LTLAXIO3:28;
       then {}l |- Q^ => (B '&&' nb) by LTLAXIO2:52,A6;
       then {}l |- 'not' (Q^) by LTLAXIO2:55;
       hence contradiction by LTLAXIO3:def 10;
     end;
     suppose
A11:   not A in rng Q`2 & not aub in rng Q`2;
       then A in rng Q`1 by XBOOLE_0:def 3,A4;
       then A12: {}l |- Q^ => A by LTLAXIO3:28;
       aub in rng Q`1 by A11,XBOOLE_0:def 3,A9;
       then {}l |- Q^ => aub by LTLAXIO3:28;
       then {}l |- Q^ => au by A12,LTLAXIO2:52;
       then {}l |- Q^ => (au '&&' ('not' au)) by LTLAXIO2:52,A7;
       then {}l |- 'not' (Q^) by LTLAXIO2:55;
       hence contradiction by LTLAXIO3:def 10;
     end;
   end;
