
theorem Th92:
  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 DJoins v9,w9,(canGFDistinction F).x9
  ex v,e,w being object st 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]
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;
  then consider G being PGraphMapping of F.x,(canGFDistinction F).x9 such that
    A2: G_V = renameElementsDistinctlyFunc(the_Vertices_of F,x) and
    A3: G_E = renameElementsDistinctlyFunc(the_Edges_of F,x) and
    A4: G is Disomorphism by Th85;
  assume A5: e9 DJoins v9,w9,(canGFDistinction F).x9;
  then A6: e9 Joins v9,w9,(canGFDistinction F).x9 by GLIB_000:16;
  A7: the_Vertices_of (canGFDistinction F).x9
     = (the_Vertices_of canGFDistinction F).x by A1, Def8
    .= [: {[the_Vertices_of F,x]}, (the_Vertices_of F).x :] by Th83
    .= rng renameElementsDistinctlyFunc(the_Vertices_of F,x) by Th80;
  x in dom F;
  then A8: x in dom the_Vertices_of F & x in dom the_Edges_of F by Def4, Def5;
  v9 in the_Vertices_of (canGFDistinction F).x9 by A6, GLIB_000:13;
  then consider v being object such that
    A9: v in (the_Vertices_of F).x & v9 = [the_Vertices_of F,x,v]
    by A7, A8, Th79;
  w9 in the_Vertices_of (canGFDistinction F).x9 by A6, GLIB_000:13;
  then consider w being object such that
    A10: w in (the_Vertices_of F).x & w9 = [the_Vertices_of F,x,w]
    by A7, A8, Th79;
  A11: the_Edges_of (canGFDistinction F).x9
     = (the_Edges_of canGFDistinction F).x by A1, Def9
    .= [: {[the_Edges_of F,x]}, (the_Edges_of F).x :] by Th84
    .= rng renameElementsDistinctlyFunc(the_Edges_of F,x) by Th80;
  e9 in the_Edges_of (canGFDistinction F).x9 by A6, GLIB_000:def 13;
  then consider e being object such that
    A12: e in (the_Edges_of F).x & e9 = [the_Edges_of F,x,e]
    by A8, A11, Th79;
  take v,e,w;
  the_Edges_of(F.x) = (the_Edges_of F).x &
    the_Vertices_of(F.x) = (the_Vertices_of F).x by Def8, Def9;
  then dom G_E = (the_Edges_of F).x & dom G_V = (the_Vertices_of F).x
    by A4, GLIB_010:def 11;
  then A13: e in dom G_E & v in dom G_V & w in dom G_V by A9, A10, A12;
  G_E.e DJoins v9,w9,(canGFDistinction F).x9 by A3, A5, A8, A12, Th78;
  then G_E.e DJoins G_V.v,w9,(canGFDistinction F).x9 by A2, A8, A9, Th78;
  then G_E.e DJoins G_V.v,G_V.w,(canGFDistinction F).x9 by A2, A8, A10, Th78;
  hence e DJoins v,w,F.x by A4, A13, GLIB_010:def 17;
  thus thesis by A9, A10, A12;
end;
