
theorem
  for n being Nat for x,y being FinSequence holds
  InnerVertices (n-BitGFA0Str(x,y)) is Relation
proof
  let n be Nat;
  let x,y be FinSequence;
  set S0 = 1GateCircStr(<*>,(0-tuples_on BOOLEAN)-->FALSE);
  deffunc S(Nat) = $1-BitGFA0Str(x,y);
  deffunc SS(Nat) = BitGFA0Str(x .($1+1), y.($1+1),
  $1-BitGFA0CarryOutput(x,y));
  defpred P[Nat] means InnerVertices (S($1)) is Relation;
  S(0) = S0 by Th2;
  then
A1: P[ 0 ] by FACIRC_1:38;
A2: now
    let i be Nat;
    assume
A3: P[i];
A4: S(i+1) = S(i) +* SS(i) by Th7;
    InnerVertices SS(i) is Relation by GFACIRC1:32;
    hence P[i+1] by A3,A4,FACIRC_1:3;
  end;
  for i being Nat holds P[i] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
