
theorem Th39:
  for x1,x2 being set, X being non empty finite set for f being
Function of 2-tuples_on X, X for S being Signature of X st x2 in the carrier of
  S & not x1 in InnerVertices S & not Output 1GateCircStr(<*x1,x2*>,f) in
  InputVertices S holds InputVertices (S +* 1GateCircStr(<*x1,x2*>,f)) = (
  InputVertices S) \/ {x1}
proof
  let x1,x2 be set, X be non empty finite set;
  set p = <*x1,x2*>;
  let f be Function of 2-tuples_on X, X;
  let S be Signature of X such that
A1: x2 in the carrier of S and
A2: not x1 in InnerVertices S;
A3: rng p = {x1,x2} by FINSEQ_2:127
    .= {x1} \/ {x2} by ENUMSET1:1;
  {x2} c= the carrier of S by A1,ZFMISC_1:31;
  hence thesis by A2,A3,Th36,ZFMISC_1:50;
end;
