
theorem Th91:
  for F being non empty Graph-yielding Function, x being Element of dom F
  for x9 being Element of dom canGFDistinction(F)
  for v,e,w being object st x = x9
  holds e Joins v,w,F.x implies [the_Edges_of F,x,e] Joins
    [the_Vertices_of F,x,v],[the_Vertices_of F,x,w],(canGFDistinction F).x9
proof
  let F be non empty Graph-yielding Function, x be Element of dom F;
  let x9 be Element of dom canGFDistinction(F), v,e,w be object;
  assume A1: x = x9;
  assume e Joins v,w,F.x;
  then per cases by GLIB_000:16;
  suppose e DJoins v,w,F.x;
    then [the_Edges_of F,x,e] DJoins [the_Vertices_of F,x,v],
      [the_Vertices_of F,x,w],(canGFDistinction F).x9 by A1, Th90;
    hence thesis by GLIB_000:16;
  end;
  suppose e DJoins w,v,F.x;
    then [the_Edges_of F,x,e] DJoins [the_Vertices_of F,x,w],
      [the_Vertices_of F,x,v],(canGFDistinction F).x9 by A1, Th90;
    hence thesis by GLIB_000:16;
  end;
end;
