
theorem Th90:
  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 DJoins v,w,F.x implies [the_Edges_of F,x,e] DJoins
    [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 x = x9;
  then consider G being PGraphMapping of F.x,(canGFDistinction F).x9 such that
    A1: G_V = renameElementsDistinctlyFunc(the_Vertices_of F,x) and
    A2: G_E = renameElementsDistinctlyFunc(the_Edges_of F,x) and
    A3: G is Disomorphism by Th85;
  assume A4: e DJoins v,w,F.x;
  then e Joins v,w,F.x by GLIB_000:16;
  then A5: e in the_Edges_of F.x & v is Vertex of F.x & w is Vertex of F.x
    by GLIB_000:13, GLIB_000:def 13;
  dom G_V = the_Vertices_of(F.x) & dom G_E = the_Edges_of(F.x)
    by A3, GLIB_010:def 11;
  then A6: G_E.e DJoins G_V.v,G_V.w,(canGFDistinction F).x9
    by A3, A4, A5, GLIB_010:def 14;
  dom F = dom the_Vertices_of F & dom F = dom the_Edges_of F by Def4, Def5;
  then A7: x in dom the_Vertices_of F & x in dom the_Edges_of F;
  (the_Vertices_of F).x = the_Vertices_of F.x by Def8;
  then A8: v in (the_Vertices_of F).x & w in (the_Vertices_of F).x by A5;
  e in (the_Edges_of F).x by A5, Def9;
  then [the_Edges_of F,x,e] DJoins G_V.v,G_V.w,(canGFDistinction F).x9
    by A2, A6, A7, Th78;
  then [the_Edges_of F,x,e] DJoins
    [the_Vertices_of F,x,v],G_V.w,(canGFDistinction F).x9
    by A1, A7, A8, Th78;
  hence thesis by A1, A7, A8, Th78;
end;
