
theorem
  for F being non empty Graph-yielding Function, x,z being Element of dom F
  for x9 being Element of dom canGFDistinction(F,z)
  for v9,e9,w9 being object
  st x <> z & x = x9 & e9 DJoins v9,w9,(canGFDistinction(F,z)).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,z be Element of dom F;
  let x9 be Element of dom canGFDistinction(F,z), v9,e9,w9 be object;
  assume A1: x <> z & x = x9 & e9 DJoins v9,w9,(canGFDistinction(F,z)).x9;
  reconsider x8 = x9 as Element of dom canGFDistinction F by FUNCT_7:30;
  (canGFDistinction(F,z)).x9 = (canGFDistinction F).x8 by A1, FUNCT_7:32;
  hence thesis by A1, Th92;
end;
