
theorem
  for x,y,z being non pair set holds InputVertices GFA3CarryStr(x,y,z)
  is without_pairs
proof
  let x,y,z be non pair set;
  set f1 = nor2, f2 = nor2, f3 = nor2, f4 = nor3;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  set Cxy = 1GateCircStr(<*x,y*>,f1);
  set Cyz = 1GateCircStr(<*y,z*>,f2);
  set Czx = 1GateCircStr(<*z,x*>,f3);
  set S = 1GateCircStr(<*xy, yz, zx*>,f4);
  set M = GFA3CarryStr(x,y,z);
  set MI = GFA3CarryIStr(x,y,z);
  given xx being pair object such that
A1: xx in InputVertices M;
A2: Cxy tolerates Cyz by CIRCCOMB:47;
A3: InnerVertices Czx = {zx} & Cxy +* Cyz tolerates Czx by CIRCCOMB:42,47;
  InnerVertices Cxy = {xy} & InnerVertices Cyz = {yz} by CIRCCOMB:42;
  then InnerVertices (Cxy +* Cyz) = {xy} \/ {yz} by A2,CIRCCOMB:11;
  then
A4: InnerVertices MI = {xy} \/ {yz} \/ {zx} by A3,CIRCCOMB:11
    .= {xy, yz} \/ {zx} by ENUMSET1:1
    .= {xy, yz, zx} by ENUMSET1:3;
  InputVertices S = {xy, yz, zx} by FACIRC_1:42;
  then
A5: InputVertices S \ InnerVertices MI = {} by A4,XBOOLE_1:37;
  InputVertices Cxy is without_pairs & InputVertices Cyz is without_pairs
  by FACIRC_1:41;
  then InputVertices Czx is without_pairs & InputVertices (Cxy+*Cyz) is
  without_pairs by FACIRC_1:9,41;
  then
A6: InputVertices MI is without_pairs by FACIRC_1:9;
  InnerVertices S is Relation by FACIRC_1:38;
  then
  InputVertices M = (InputVertices MI) \/ (InputVertices S \ InnerVertices
  MI) by A6,FACIRC_1:6;
  hence thesis by A6,A1,A5;
end;
