reserve A,B,p,q,r,s for Element of LTLB_WFF,
  n for Element of NAT,
  X for Subset of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y for set;
reserve P,Q,P1,R for PNPair;

theorem Th28: A in rng P`1 implies {} LTLB_WFF |- (P^) => A
  proof
    set fp = P`1,fm = P`2,nfm = nega fm;
    assume A in rng fp;
    then consider i being Nat such that
A1: i in dom fp and
A2: fp . i = A by FINSEQ_2:10;
    (P^) => A is ctaut
    proof
      let g;
      set v = VAL g,r = v.kon(fp|(i -' 1)),s = v.kon(fp/^i);
A3:   v.A = 1 or v.A = 0 by XBOOLEAN:def 3;
      thus v.((P^) => A) = v.(P^) => v.A by LTLAXIO1:def 15
      .= (v.kon(fp) '&' v.kon(nfm)) => v.A by LTLAXIO1:31
      .= (r '&' v.(fp/.i) '&' s '&' v.kon(nfm)) => v.A by LTLAXIO2:18, A1
      .= ((r '&' v.A '&' s) '&' v.kon(nfm)) => v.A by PARTFUN1:def 6, A1, A2
      .= 1 by A3;
    end;
    then (P^) => A in LTL_axioms by LTLAXIO1:def 17;
    hence {} l |- (P^) => A by LTLAXIO1:42;
  end;
