reserve A,B,p,q,r,s for Element of LTLB_WFF,
  i,j,k,n for Element of NAT,
  X for Subset of LTLB_WFF,
  f,f1 for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN;

theorem
  rng f c= rng f1 implies ('not' ((con (nega f)) /. (len (con (nega f))))) =>
  ('not' ((con (nega f1)) /. (len (con (nega f1))))) is ctaut
  proof
    assume
A1: rng f c= rng f1;
    set p = alt(f) => alt(f1);
    assume
    not p is ctaut;
    then consider g such that
A2: not (VAL g).p = 1;
    set v = VAL g;
    v.p = 0 by A2,XBOOLEAN:def 3;
    then A3: v.alt(f) => v.alt(f1) = 0 by LTLAXIO1:def 15;
A4: v.alt(f) = TRUE or v.alt(f) = FALSE by XBOOLEAN:def 3;
    now
      let i be Nat;
      assume
A5:   i in dom f;
      then f.i in rng f by FUNCT_1:3;
      then consider j be object such that
A6:   j in dom f1 and
A7:   f.i = f1.j by A1,FUNCT_1:def 3;
      f1/.j = f1.j by PARTFUN1:def 6,A6
      .= f/.i by A5,PARTFUN1:def 6,A7;
      hence v.(f/.i) = 0 by A6,Th20, A3,A4;
      reconsider j as Nat by A6;
    end;
    hence contradiction by A3,A4,Th20;
  end;
