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 for P be consistent PNPair holds rng P`1 misses rng P`2
  proof
    let P be consistent PNPair;
    assume not rng P`1 misses rng P`2;
    then not rng P`1 /\ rng P`2 = {};
    then consider x being object such that
A1: x in rng P`1 /\ rng P`2 by XBOOLE_0:def 1;
    x in rng P`1 by XBOOLE_0:def 4,A1;
    then consider n1 be object such that
A2: n1 in dom P`1 and
A3: P`1.n1 = x by FUNCT_1:def 3;
    x in rng P`2 by XBOOLE_0:def 4,A1;
    then consider n2 be object such that
A4: n2 in dom P`2 and
A5: P`2.n2 = x by FUNCT_1:def 3;
    reconsider n1,n2 as Element of NAT by A2,A4;
    x = P`2/.n2 by PARTFUN1:def 6,A4,A5;
    then reconsider x as Element of l;
A6: 1<=n2 by FINSEQ_3:25,A4;
A7: n2 <= len P`2 by FINSEQ_3:25,A4;
    'not' (P^) is ctaut
    proof
      let g;
      set nP2 = nega P`2,v = VAL g;
A8:   v.x = TRUE or v.x = FALSE by XBOOLEAN:def 3;
A9:   v.(P`1/.n1) = v.x by PARTFUN1:def 6,A2,A3;
      n2 <= len nP2 by A7,LTLAXIO2:def 4;
      then A10: n2 in dom nP2 by FINSEQ_3:25,A6;
A11:  v.(nP2/.n2) = v.('not' ((P`2)/.n2)) by LTLAXIO2:8,A4
      .= v.('not' x) by A4,A5,PARTFUN1:def 6
      .= v.x => v.TFALSUM by LTLAXIO1:def 15
      .= v.x => FALSE by LTLAXIO1:def 15;
A12:  v.(P^) = v.kon(P`1) '&' v.kon(nega P`2) by LTLAXIO1:31
      .= v.kon(P`1|(n1-'1)) '&' v.(P`1/.n1) '&' v.kon(P`1/^n1)'&'v.kon(nP2)
      by LTLAXIO2:18,A2
      .= v.kon(P`1|(n1-'1)) '&' v.(P`1/.n1) '&' v.kon(P`1/^n1) '&'
    (v.kon(nP2|(n2-'1)) '&' v.(nP2/.n2) '&' v.kon(nP2/^n2)) by LTLAXIO2:18,A10
      .= 0 by A8,A11,A9;
      thus v.('not' (P^)) = v.(P^) => v.TFALSUM by LTLAXIO1:def 15
      .= 1 by A12;
    end;
    then 'not' (P^) in LTL_axioms by LTLAXIO1:def 17;
    then {} l |- 'not' (P^) by LTLAXIO1:42;
    hence contradiction by Def10;
  end;
