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 nega (f^<*A*>) = (nega f)^<*'not' A*>
  proof
    set p = nega (f^<*A*>),q = (nega f)^<*'not' A*>;
    len p = len (f^<*A*>) by Def4;
    then A1: dom p = dom (f^<*A*>) by FINSEQ_3:29;
A2: len p = len (f^<*A*>) by Def4
    .= len f + len <*A*> by FINSEQ_1:22
    .= len f + 1 by FINSEQ_1:39
    .= len nega f + 1 by Def4
    .= len nega f + len <*'not' A*> by FINSEQ_1:39
    .= len q by FINSEQ_1:22;
    now
      let j be Nat;
A3:   len (f^<*A*>) = len f + len <*A*> by FINSEQ_1:22
      .= len f + 1 by FINSEQ_1:39;
      assume
A4:   j in dom p;
      then A5: 1 <= j by FINSEQ_3:25;
      j <= len p by A4,FINSEQ_3:25;
      then A6: j <= len f + 1 by A3,Def4;
A7:   j in dom q by A4, A2,FINSEQ_3:29;
      per cases by A6,XXREAL_0:1;
      suppose
A8:     j = len f + 1;
        then A9: j = len (nega f) + 1 by Def4;
        thus p.j = p/.j by PARTFUN1:def 6,A4
        .= 'not' (f^<*A*>)/.j by Th8,A4,A1
        .= 'not' A by FINSEQ_4:67,A8
        .= q/.j by A9,FINSEQ_4:67
        .= q.j by PARTFUN1:def 6,A7;
      end;
      suppose
        j < len f + 1;
        then A10: j <= len f by NAT_1:13;
        then A11: j in dom f by A5,FINSEQ_3:25;
        j <= len nega f by A10,Def4;
        then A12: j in dom nega f by FINSEQ_3:25,A5;
        thus p.j = p/.j by PARTFUN1:def 6,A4
        .= 'not' (f^<*A*>)/.j by Th8,A4,A1
        .= 'not' f/.j by FINSEQ_4:68,A11
        .= (nega f)/.j by Th8,A11
        .= q/.j by FINSEQ_4:68,A12
        .= q.j by PARTFUN1:def 6,A7;
      end;
    end;
    hence thesis by FINSEQ_2:9,A2;
  end;
