
theorem
  for F being non empty Graph-yielding Function, z being Element of dom F
  for z9 being Element of dom canGFDistinction(F,z)
  for v,e,w being object st z = z9
  holds e DJoins v,w,F.z iff e DJoins v,w,canGFDistinction(F,z).z9
proof
  let F be non empty Graph-yielding Function, z be Element of dom F;
  let z9 be Element of dom canGFDistinction(F,z), v,e,w be object;
  assume A1: z = z9;
  F.z == F.z | _GraphSelectors by GLIB_000:128;
  then F.z == canGFDistinction(F,z).z9 by A1, Th96;
  hence thesis by GLIB_000:88;
end;
