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 Th29: A in rng P`2 implies {} LTLB_WFF |- (P^) => 'not' A
  proof
    set fp = P`1,fm = P`2,nfm = nega fm;
    assume A in rng fm;
    then consider i being Nat such that
A1: i in dom fm and
A2: fm . i = A by FINSEQ_2:10;
    reconsider i as Element of NAT by ORDINAL1:def 12;
    i <= len fm by A1,FINSEQ_3:25;
    then A3: i <= len nfm by LTLAXIO2:def 4;
    1 <= i by A1,FINSEQ_3:25;
    then A4: i in dom nfm by A3,FINSEQ_3:25;
    (P^) => 'not' A is ctaut
    proof
      let g;
      set v = VAL g,p = v.kon(fp),q = v.kon(nfm), r = v.kon(nfm|(i -' 1)),
      s = v.kon(nfm/^i);
A5:   v.('not' A) = 1 or v.('not' A) = 0 by XBOOLEAN:def 3;
A6:   q = r '&' v.(nfm/.i) '&' s by LTLAXIO2:18,A4
      .= r '&' v.('not' (fm/.i)) '&' s by LTLAXIO2:8,A1
      .= r '&' v.('not' A) '&' s by PARTFUN1:def 6,A1,A2;
      thus v.((P^) => ('not' A)) = v.(P^) => v.('not' A) by LTLAXIO1:def 15
      .= (p '&' q) => v.('not' A) by LTLAXIO1:31
      .= 1 by A5,A6;
    end;
    then (P^) => 'not' A in LTL_axioms by LTLAXIO1:def 17;
    hence {} l |- (P^) => ('not' A) by LTLAXIO1:42;
  end;
