
theorem Th104:
  for F being non empty Graph-yielding Function, x,z being Element of dom F
  for x9 being Element of dom canGFDistinction(F,z) st x = x9
  holds (canGFDistinction(F,z)).x9 is F.x-Disomorphic
proof
  let F be non empty Graph-yielding Function, x,z be Element of dom F;
  let x9 be Element of dom canGFDistinction(F,z);
  assume A1: x = x9;
  per cases;
  suppose x = z;
    then canGFDistinction(F,z).x = F.x | _GraphSelectors by Th96;
    hence thesis by A1, GLIB_000:128, GLIBPRE0:76;
  end;
  suppose x <> z;
    then consider G being PGraphMapping of F.x,(canGFDistinction(F,z)).x9
      such that G_V = renameElementsDistinctlyFunc(the_Vertices_of F,x) &
      G_E = renameElementsDistinctlyFunc(the_Edges_of F,x) and
      A2: G is Disomorphism by A1, Th103;
    thus thesis by A2, GLIB_010:def 24;
  end;
end;
