
theorem Th51:
  for x1 being set, X being non empty finite set for f being
Function of 1-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 holds S +* 1GateCircStr(<*x1*>, f) is
  with_nonpair_inputs
proof
  let x1 be set, X be non empty finite set;
  let f be Function of 1-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;
A2: not Output 1GateCircStr(<*x1*>, f) in InputVertices S by FACIRC_1:def 2;
  per cases by A1;
  suppose
    x1 in the carrier of S;
    hence InputVertices (S +* 1GateCircStr(<*x1*>, f)) is without_pairs by A2
,Th37;
  end;
  suppose
    x1 is non pair;
    then reconsider a = x1 as non pair set;
    rng <*x1*> = {a} by FINSEQ_1:38;
    hence InputVertices (S +* 1GateCircStr(<*x1*>, f)) is without_pairs;
  end;
end;
