
theorem Th41:
  for x1,x2,x3 being set, X being non empty finite set for f being
Function of 3-tuples_on X, X for S being Signature of X st x1 in the carrier of
  S & not x2 in InnerVertices S & not x3 in InnerVertices S & not Output
  1GateCircStr(<*x1,x2,x3*>,f) in InputVertices S holds InputVertices (S +*
  1GateCircStr(<*x1,x2,x3*>,f)) = (InputVertices S) \/ {x2,x3}
proof
  let x1,x2,x3 be set, X be non empty finite set;
  set p = <*x1,x2,x3*>;
  let f be Function of 3-tuples_on X, X;
  let S be Signature of X such that
A1: x1 in the carrier of S and
A2: ( not x2 in InnerVertices S)& not x3 in InnerVertices S;
A3: rng p = {x1,x2,x3} by FINSEQ_2:128
    .= {x1} \/ {x2,x3} by ENUMSET1:2;
  {x1} c= the carrier of S by A1,ZFMISC_1:31;
  hence thesis by A2,A3,Th36,ZFMISC_1:51;
end;
