
theorem Th36:
  for X1,X2 being set, X being non empty finite set, n be Element
of NAT 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 = X1 \/ X2 & X1 c= the carrier of S & X2 misses
  InnerVertices S & not Output 1GateCircStr(p,f) in InputVertices S holds
  InputVertices (S +* 1GateCircStr(p,f)) = (InputVertices S) \/ X2
proof
  let x1,x2 be set, X be non empty finite set, n be Element of NAT;
  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 = x1 \/ x2 and
A2: x1 c= the carrier of S and
A3: x2 misses InnerVertices S and
A4: not Output 1GateCircStr(p,f) in InputVertices S;
A5: 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)\(InnerVertices 1GateCircStr(p,f))) \/ ((x1\(
  InnerVertices S)) \/ x2) by A1,A3,XBOOLE_1:87
    .= ((InputVertices S)\(InnerVertices 1GateCircStr(p,f))) \/ (x1\(
  InnerVertices S)) \/ x2 by XBOOLE_1:4
    .= ((InputVertices S) \ {Output 1GateCircStr(p,f)}) \/ (x1\(
  InnerVertices S)) \/ x2 by Th16
    .= (InputVertices S) \/ (x1\(InnerVertices S)) \/ x2 by A4,ZFMISC_1:57
    .= (InputVertices S) \/ x2 by A2,A5,XBOOLE_1:12,43;
end;
