
theorem Th53:
  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 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) & (x3 in the carrier of S or x3 is non pair) holds S +* 1GateCircStr(
  <*x1,x2,x3*>, f) is with_nonpair_inputs
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 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) &( x3 in the carrier of S or x3 is non pair);
A2: not Output 1GateCircStr(<*x1,x2,x3*>, 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 & x3 in the
carrier of S or x1 in the carrier of S & not x2 in InnerVertices S & x3 in the
carrier of S or x2 in the carrier of S & not x1 in InnerVertices S & x3 in the
    carrier of S or x1 in the carrier of S & x2 in the carrier of S & not x3 in
InnerVertices S or x1 in the carrier of S & not x2 in InnerVertices S & not x3
in InnerVertices S or x2 in the carrier of S & not x1 in InnerVertices S & not
x3 in InnerVertices S or not x1 in InnerVertices S & not x2 in InnerVertices S
    & x3 in the carrier of S;
    then InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) = InputVertices S
or {x2} is without_pairs & InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) =
    {x2} \/ InputVertices S or {x1} is without_pairs & InputVertices (S +*
    1GateCircStr(<*x1,x2,x3*>, f)) = {x1} \/ InputVertices S or {x3} is
without_pairs & InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) = {x3} \/
InputVertices S or {x1,x2} is without_pairs & InputVertices (S +* 1GateCircStr(
<*x1,x2,x3*>, f)) = {x1,x2} \/ InputVertices S or {x2,x3} is without_pairs &
InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) = {x2,x3} \/ InputVertices S
or {x1,x3} is without_pairs & InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)
    ) = {x1,x3} \/ InputVertices S by A1,A2,Th41,Th42,Th43,Th44,Th45,Th46,Th47;
    hence InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) is without_pairs;
  end;
  suppose
    x1 is non pair & x2 is non pair & x3 is non pair;
    then reconsider a = x1, b = x2, c = x3 as non pair set;
    rng <*x1,x2,x3*> = {a,b,c} by FINSEQ_2:128;
    hence InputVertices (S +* 1GateCircStr(<*x1,x2,x3*>, f)) is without_pairs;
  end;
end;
