
theorem Th69:
  for x,y,z being set holds [<*x,y*>,xor2c] in InnerVertices
BitGFA1Str(x,y,z) & GFA1AdderOutput(x,y,z) in InnerVertices BitGFA1Str(x,y,z) &
  [<*x,y*>,and2c] in InnerVertices BitGFA1Str(x,y,z) & [<*y,z*>,and2a] in
InnerVertices BitGFA1Str(x,y,z) & [<*z,x*>,and2] in InnerVertices BitGFA1Str(x,
  y,z) & GFA1CarryOutput(x,y,z) in InnerVertices BitGFA1Str(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 A1 = GFA1AdderOutput(x,y,z);
  set A2 = GFA1CarryOutput(x,y,z);
  InnerVertices S = {xyf0} \/ {A1} \/ {xyf1,yzf2,zxf3} \/ {A2} by Th63
    .= {xyf0,A1} \/ {xyf1,yzf2,zxf3} \/ {A2} by ENUMSET1:1
    .= {xyf0,A1,xyf1,yzf2,zxf3} \/ {A2} by ENUMSET1:8
    .= {xyf0,A1,xyf1,yzf2,zxf3,A2} by ENUMSET1:15;
  hence thesis by ENUMSET1:def 4;
end;
