
theorem Th35:
  for n being Element of NAT, X being non empty finite set for f
  being Function of n-tuples_on X, X for p being FinSeqLen of n for S being
  Signature of X st rng p c= the carrier of S & not Output 1GateCircStr(p,f) in
InputVertices S holds InputVertices (S +* 1GateCircStr(p,f)) = InputVertices S
proof
  let n be Element of NAT, X be non empty finite set;
  let f be Function of n-tuples_on X, X;
  let p be FinSeqLen of n;
  let S be Signature of X such that
A1: rng p c= the carrier of S and
A2: not Output 1GateCircStr(p,f) in InputVertices S;
A3: the carrier of S = (InputVertices S) \/ InnerVertices S by XBOOLE_1:45;
  thus InputVertices (S +* 1GateCircStr(p,f)) = ((InputVertices S)\(
  InnerVertices 1GateCircStr(p,f))) \/ ((InputVertices 1GateCircStr(p,f))\(
  InnerVertices S)) by CIRCCMB2:5,CIRCCOMB:47
    .= ((InputVertices S)\(InnerVertices 1GateCircStr(p,f))) \/ ((rng p)\(
  InnerVertices S)) by CIRCCOMB:42
    .= ((InputVertices S) \ {Output 1GateCircStr(p,f)}) \/ ((rng p)\(
  InnerVertices S)) by Th16
    .= (InputVertices S) \/ ((rng p)\(InnerVertices S)) by A2,ZFMISC_1:57
    .= InputVertices S by A1,A3,XBOOLE_1:12,43;
end;
