
theorem Th47:
  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 & x2 in the carrier of S & x3 in the carrier of S & not Output 1GateCircStr(
<*x1,x2,x3*>,f) in InputVertices S holds InputVertices (S +* 1GateCircStr(<*x1,
  x2,x3*>,f)) = InputVertices S
proof
  let x1,x2,x3 be set, X be non empty finite set;
  let f be Function of 3-tuples_on X, X;
  let S be Signature of X;
  assume x1 in the carrier of S & x2 in the carrier of S & x3 in the carrier
  of S; then
A1: {x1,x2} c= the carrier of S & {x3} c= the carrier of S by ZFMISC_1:31,32;
  rng <*x1,x2,x3*> = {x1,x2,x3} by FINSEQ_2:128
    .= {x1,x2} \/ {x3} by ENUMSET1:3;
  hence thesis by A1,Th35,XBOOLE_1:8;
end;
