
theorem Th52:
  for x1,x2 being set, X being non empty finite set for f being
Function of 2-tuples_on X, X for S being with_nonpair_inputs Signature of X st
  (x1 in the carrier of S or x1 is non pair) & (x2 in the carrier of S or x2 is
  non pair) holds S +* 1GateCircStr(<*x1,x2*>, f) is with_nonpair_inputs
proof
  let x1,x2 be set, X be non empty finite set;
  let f be Function of 2-tuples_on X, X;
  let S be with_nonpair_inputs Signature of X such that
A1: ( x1 in the carrier of S or x1 is non pair)&( x2 in the carrier of S
  or x2 is non pair);
A2: not Output 1GateCircStr(<*x1,x2*>, f) in InputVertices S by FACIRC_1:def 2;
  per cases by A1;
  suppose
    x1 in the carrier of S & x2 in the carrier of S or x1 in the
carrier of S & not x2 in InnerVertices S or x2 in the carrier of S & not x1 in
    InnerVertices S;
    then
    InputVertices (S +* 1GateCircStr(<*x1,x2*>, f)) = InputVertices S or {
x2} is without_pairs & InputVertices (S +* 1GateCircStr(<*x1,x2*>, f)) = {x2}
\/ InputVertices S or {x1} is without_pairs & InputVertices (S +* 1GateCircStr(
    <*x1,x2*>, f)) = {x1} \/ InputVertices S by A1,A2,Th38,Th39,Th40;
    hence InputVertices (S +* 1GateCircStr(<*x1,x2*>, f)) is without_pairs;
  end;
  suppose
    x1 is non pair & x2 is non pair;
    then reconsider a = x1, b = x2 as non pair set;
    rng <*x1,x2*> = {a,b} by FINSEQ_2:127;
    hence InputVertices (S +* 1GateCircStr(<*x1,x2*>, f)) is without_pairs;
  end;
end;
