reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th33:
  A is LTL_TAUT_OF_PL implies F|=A
 proof
  assume A1: A is LTL_TAUT_OF_PL;
  let M be LTLModel;
  assume M|=F;
  let n;
  defpred P[object,object] means
   $2=(SAT M).[n,$1];
  A2: for x be object st x in LTLB_WFF
ex y be object st y in BOOLEAN & P[x,y]
  proof
   let x be object;
   set y=(SAT M).[n,x];
   assume x in LTLB_WFF;
   then reconsider x1=x as Element of LTLB_WFF;
   take y;
   (SAT M).[n,x1] in BOOLEAN;
   hence y in BOOLEAN & P[x,y];
  end;
  consider f be Function of LTLB_WFF,BOOLEAN such that
   A3: for B be object st B in LTLB_WFF holds P[B,f.B]
from FUNCT_2:sch 1(A2);
  thus(SAT M).[n,A]=(VAL f).A by A3,Th32
   .=1 by A1;
 end;
