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 Th28: R in rng T & Q in untn R implies comp Q c= rngr T
  proof
    assume that
A1: R in rng T and
A2: Q in untn R;
    let S be object;
    assume
A3: S in comp Q;
    comp Q c= comp untn R by Th14,A2;
    then A4: S in compn R by A3;
    consider t be object such that
A5: t in dom T and
A6: T.t = R by FUNCT_1:def 3,A1;
    reconsider t as Element of dom T by A5;
    succ (T,t) = the Enumeration of compn R by Def11,A6;
    then S in rng succ (T,t) by RLAFFIN3:def 1,A4;
    then consider t1 be Element of dom T such that
A7: S = T.t1 and
A8: t1 in succ t by TREES_9:42;
    ex n being Nat st t1 = t^<*n*> & t^<*n*> in dom T by A8;
    hence S in rngr T by A7;
  end;
