
theorem Th10:
  for x,y,z being set holds InnerVertices GFA0CarryIStr(x,y,z) = {
  [<*x,y*>,and2], [<*y,z*>,and2], [<*z,x*>,and2]}
proof
  let x,y,z be set;
  set f1 = and2, f2 = and2, f3 = and2;
  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);
A1: Cxy tolerates Cyz by CIRCCOMB:47;
  Cxy +* Cyz tolerates Czx by CIRCCOMB:47;
  then InnerVertices GFA0CarryIStr(x,y,z) = InnerVertices(Cxy +* Cyz) \/
  InnerVertices(Czx) by CIRCCOMB:11
    .= InnerVertices(Cxy) \/ InnerVertices(Cyz) \/ InnerVertices(Czx) by A1,
CIRCCOMB:11
    .= {xy} \/ InnerVertices(Cyz) \/ InnerVertices(Czx) by CIRCCOMB:42
    .= {xy} \/ {yz} \/ InnerVertices(Czx) by CIRCCOMB:42
    .= {xy} \/ {yz} \/ {zx} by CIRCCOMB:42
    .= {xy, yz} \/ {zx} by ENUMSET1:1
    .= {xy, yz, zx} by ENUMSET1:3;
  hence thesis;
end;
