
theorem Th93:
  for F being non empty Graph-yielding Function, x being Element of dom F
  for x9 being Element of dom canGFDistinction(F)
  for v9,e9,w9 being object st x = x9 & e9 Joins v9,w9,(canGFDistinction F).x9
  ex v,e,w being object st e Joins v,w,F.x &
    e9 = [the_Edges_of F,x,e] & v9 = [the_Vertices_of F,x,v] &
    w9 = [the_Vertices_of F,x,w]
proof
  let F be non empty Graph-yielding Function, x be Element of dom F;
  let x9 be Element of dom canGFDistinction(F), v9,e9,w9 be object;
  assume A1: x = x9 & e9 Joins v9,w9,(canGFDistinction F).x9;
  then per cases by GLIB_000:16;
  suppose e9 DJoins v9,w9,(canGFDistinction F).x9;
    then consider v,e,w being object such that
      A2: e DJoins v,w,F.x & e9 = [the_Edges_of F,x,e] &
        v9 =[the_Vertices_of F,x,v] & w9 =[the_Vertices_of F,x,w] by A1, Th92;
    take v,e,w;
    thus thesis by A2, GLIB_000:16;
  end;
  suppose e9 DJoins w9,v9,(canGFDistinction F).x9;
    then consider w,e,v being object such that
      A3: e DJoins w,v,F.x & e9 = [the_Edges_of F,x,e] &
        w9 =[the_Vertices_of F,x,w] & v9 =[the_Vertices_of F,x,v] by A1, Th92;
    take v,e,w;
    thus thesis by A3, GLIB_000:16;
  end;
end;
