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 Th23:
  R in compn Q & rng Q c= union Subt rng P implies rng R c= union Subt rng P
  proof
    assume that
A1: R in compn Q and
A2: rng Q c= union Subt rng P;
    consider R1 be PNPair such that
A3: R1 = R and
A4: R1 in comp untn Q by A1;
    consider z such that
A5: R1 in z and
A6: z in {comp S where S is PNPair: S in untn Q} by A4,TARSKI:def 4;
    consider T be PNPair such that
A7: z = comp T and
A8: T in untn Q by A6;
A9: ex T1 be PNPair st T1 = T & rng T1 `1 = untn rng Q`1 &
    rng T1`2 = untn rng Q`2 by A8;
    let x be object;
    assume
A10: x in rng R;
     ex R2 be consistent PNPair st R2=R1 & R2 is_completion_of T by A7, A5;
     then x in tau rng T by A3,A10;
     then consider p such that
A11: p in rng T and
A12: x in tau1.p by LTLAXIO3:def 5;
     per cases by XBOOLE_0:def 3, A9,A11;
     suppose p in untn rng Q`1;
       then consider A,B such that
A13:   p = untn(A,B) and
A14:   A 'U' B in rng (Q`1);
       set aub = A 'U' B;
       rng (Q`1) c= rng Q by XBOOLE_1:7;
       then aub in rng Q by A14;
       then consider y such that
A15:   aub in y and
A16:   y in Subt rng P by A2,TARSKI:def 4;
       consider q be Element of l such that
A17:   y = Sub.q and
       q in rng P by A16;
       tau1.untn(A,B) c= tau1.untn(A,B) \/ (Sub.A \/ Sub.B) by XBOOLE_1:7;then
       tau1.untn(A,B) c= tau1.untn(A,B) \/ Sub.A \/ Sub.B by XBOOLE_1:4;
       then tau1.untn(A,B) c= Sub.aub by LTLAXIO3:def 6;
       then x in Sub.q by A17,A15,LTLAXIO3:26, A13,A12;
       hence x in union Subt rng P by A17,A16,TARSKI:def 4;
     end;
     suppose p in untn rng Q`2;
       then consider A,B such that
A18:   p = untn(A,B) and
A19:   A 'U' B in rng (Q`2);
       set aub = A 'U' B;
       rng (Q`2) c= rng Q by XBOOLE_1:7;
       then aub in rng Q by A19;
       then consider y such that
A20:   aub in y and
A21:   y in Subt rng P by A2,TARSKI:def 4;
       consider q be Element of l such that
A22:   y = Sub.q and
       q in rng P by A21;
       tau1.untn(A,B) c= tau1.untn(A,B) \/ (Sub.A \/ Sub.B) by XBOOLE_1:7;
       then tau1.untn(A,B) c= tau1.untn(A,B) \/ Sub.A \/ Sub.B by XBOOLE_1:4;
       then tau1.untn(A,B) c= Sub.aub by LTLAXIO3:def 6;
       then x in Sub.q by A22,A20,LTLAXIO3:26, A18,A12;
       hence x in union Subt rng P by A22,A21,TARSKI:def 4;
     end;
   end;
