
theorem Th63:
  for x,y,z being set holds InnerVertices BitGFA1Str(x,y,z) = {[<*
x,y*>,xor2c]} \/ {GFA1AdderOutput(x,y,z)} \/ {[<*x,y*>,and2c], [<*y,z*>,and2a],
  [<*z,x*>,and2]} \/ {GFA1CarryOutput(x,y,z)}
proof
  let x,y,z be set;
  set f1 = and2c, f2 = and2a, f3 = and2;
  set f0 = xor2c;
  set xyf0 = [<*x,y*>,f0];
  set xyf1 = [<*x,y*>,f1], yzf2 = [<*y,z*>,f2], zxf3 = [<*z,x*>,f3];
  set S = BitGFA1Str(x,y,z);
  set S1 = GFA1AdderStr(x,y,z);
  set S2 = GFA1CarryStr(x,y,z);
  set A1 = GFA1AdderOutput(x,y,z);
  set A2 = GFA1CarryOutput(x,y,z);
  thus InnerVertices S = (InnerVertices S1) \/ InnerVertices S2 by FACIRC_1:27
    .= {xyf0} \/ {A1} \/ InnerVertices S2 by Th55
    .= {xyf0} \/ {A1} \/ ({xyf1, yzf2, zxf3} \/ {A2}) by Th42
    .= {xyf0} \/ {A1} \/ {xyf1, yzf2, zxf3} \/ {A2} by XBOOLE_1:4;
end;
