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 Th27: Q in rng T implies rng Q c= union Subt rng P
  proof
    deffunc F(PNPair) = {Sub.A where A is Element of l: A in rng $1};
    defpred P1[set] means
    for Q st Q = T.$1 & $1 in dom T holds rng Q c= union Subt rng P;
    assume Q in rng T;
    then consider x being object such that
A1: x in dom T and
A2: T.x = Q by FUNCT_1:def 3;
    reconsider x as Element of dom T by A1;
A3: now
      let t be Element of dom T,n be Nat;
      assume that
A4:   P1[t] and
      t^<*n*> in dom T;
A5:   rng (T.t) c= union Subt rng P by A4;
      thus P1[t^<*n*>]
      proof
        let Q;
        assume Q = T.(t^<*n*>) & t^<*n*> in dom T;
        then Q in compn (T.t) by Th26;
       hence rng Q c= union Subt rng P by Th23,A5;
     end;
   end;
A6: P1[{}]
    proof
      let Q;
      assume that
A7:   Q = T.{} and {} in dom T;
      Q = P by A7,Def11;
      hence thesis by Th9;
    end;
    for t be Element of dom T holds P1[t] from TREES_2:sch 1(A6,A3 );
    then rng (T.x) c= union Subt rng P;
    hence thesis by A2;
  end;
