
theorem Th109:
  for x,y,z being set st x <> [<*y,z*>,nor2] & y <> [<*z,x*>,
nor2] & z <> [<*x,y*>,nor2] holds InputVertices GFA3CarryStr(x,y,z) = {x,y,z}
proof
  let x,y,z be set;
  set f1 = nor2, f2 = nor2, f3 = nor2, f4 = nor3;
  set xy = [<*x,y*>,f1], yz = [<*y,z*>,f2], zx = [<*z,x*>,f3];
  set xyz = [<*xy, yz, zx*>,f4];
  set S = 1GateCircStr(<*xy, yz, zx*>,f4);
  set MI = GFA3CarryIStr(x,y,z);
A1: InputVertices S = rng <*xy, yz, zx*> by CIRCCOMB:42
    .= {xy, yz, zx} by FINSEQ_2:128;
  assume
A2: x <> yz & y <> zx & z <> xy;
A3: InnerVertices S = {xyz} & {x, y, z} \ {xyz} = {x, y, z} by Lm2,CIRCCOMB:42;
A4: {xy, yz, zx} \ {xy, yz, zx} = {} by XBOOLE_1:37;
  thus InputVertices GFA3CarryStr(x,y,z) = ((InputVertices MI) \ InnerVertices
  S) \/ ((InputVertices S) \ InnerVertices MI) by CIRCCMB2:5,CIRCCOMB:47
    .= {x,y,z} \/ ({xy, yz, zx} \ InnerVertices MI) by A1,A2,A3,Th108
    .= {x,y,z} \/ {} by A4,Th105
    .= {x,y,z};
end;
