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 = rng f1 implies
   (VAL g).((con f)/.(len con f)) = (VAL g).((con f1)/.(len con f1))
   proof
     set v = VAL g;
     assume
A1:  rng f = rng f1;
     per cases;
     suppose
A2:    len f = 0;
       then len f1 = 0 by Th3,A1;
       then A3: f1 = {};
       f = {} by A2;
       hence v.kon(f) = v.kon(f1) by A3;
     end;
     suppose
       len f > 0;
       per cases by XBOOLEAN:def 3;
       suppose
A4:      v.kon(f) = 1;
         assume
         not v.kon(f) = v.kon(f1);
         then consider i be Nat such that
A5:      i in dom f1 and
A6:      not v.(f1/.i) = 1 by A4,Th19;
         set j = f1/.i .. f;
         f1/.i in rng f by A5,PARTFUN2:2,A1;then
         j in dom f & v.(f/.j) = v.(f1/.i) by FINSEQ_4:20, FINSEQ_5:38;
         hence contradiction by Th19,A4, A6;
       end;
       suppose
A7:      v.kon(f) = 0;
         assume
A8:      not v.kon(f) = v.kon(f1);
         consider i be Nat such that
A9:      i in dom f and
A10:     not v.(f/.i) = 1 by A7,Th19;
         set j = f/.i .. f1;
A11:     f/.i in rng f1 by A9,PARTFUN2:2,A1;
         then j in dom f1 by FINSEQ_4:20;
         then v.(f1/.j) = 1 by Th19, A8,A7,XBOOLEAN:def 3;
         hence contradiction by A10, FINSEQ_5:38,A11;
       end;
     end;
   end;
