
theorem
  for n being Element of NAT for x,y being FinSequence holds
  InnerVertices (n-BitSubtracterStr(x,y)) is Relation
proof
  let n be Element of NAT;
  let x,y be FinSequence;
  defpred P[Nat] means
  InnerVertices ($1-BitSubtracterStr(x,y)) is Relation;
  0-BitSubtracterStr(x,y) = 1GateCircStr(<*>,(0-tuples_on BOOLEAN)--> TRUE )
  by Th2;
  then
A1: P[ 0 ] by FACIRC_1:38;
A2: now
    let i be Nat;
    assume
A3: P[i];
A4: (i+1)-BitSubtracterStr(x, y) = i-BitSubtracterStr(x, y) +*
    BitSubtracterWithBorrowStr(x .(i+1), y.(i+1),
    i-BitBorrowOutput(x, y)) by Th7;
    InnerVertices BitSubtracterWithBorrowStr(x .(i+1), y.(i+1),
    i-BitBorrowOutput(x, y)) is Relation by FSCIRC_1:22;
    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;
