
theorem
  for n being Element of NAT for x,y being FinSequence holds
  InnerVertices (n-BitAdderStr(x,y)) is Relation
proof
  let n be Element of NAT;
  let x,y be FinSequence;
  defpred P[Nat] means
  InnerVertices ($1-BitAdderStr(x,y)) is Relation;
  0-BitAdderStr(x,y) = 1GateCircStr(<*>,(0-tuples_on BOOLEAN)-->FALSE) by Th7;
  then
A1: P[ 0 ] by FACIRC_1:38;
A2: for i being Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume
A3: InnerVertices (i-BitAdderStr(x,y)) is Relation;
A4: (i+1)-BitAdderStr(x, y) = i-BitAdderStr(x, y) +*
    BitAdderWithOverflowStr(x .(i+1), y.(i+1),
    i-BitMajorityOutput(x, y)) by Th12;
    InnerVertices BitAdderWithOverflowStr(x .(i+1), y.(i+1),
    i-BitMajorityOutput(x, y)) is Relation by FACIRC_1:88;
    hence thesis by A3,A4,FACIRC_1:3;
  end;
  for i being Nat holds P[i] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
