
theorem Th103:
  for F being non empty Graph-yielding Function, x,z being Element of dom F
  for x9 being Element of dom canGFDistinction(F,z) st x <> z & x = x9
  ex G being PGraphMapping of F.x,(canGFDistinction(F,z)).x9
  st G_V = renameElementsDistinctlyFunc(the_Vertices_of F,x) &
    G_E = renameElementsDistinctlyFunc(the_Edges_of F,x) &
    G is Disomorphism
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);
  assume A1: x <> z & x = x9;
  reconsider x8 = x9 as Element of dom canGFDistinction F by FUNCT_7:30;
  consider G0 being PGraphMapping of F.x, (canGFDistinction F).x8 such that
    A2: G0_V = renameElementsDistinctlyFunc(the_Vertices_of F,x) &
      G0_E = renameElementsDistinctlyFunc(the_Edges_of F,x) and
    A3: G0 is Disomorphism by A1, Th85;
  A4: (canGFDistinction F).x8 = canGFDistinction(F,z).x9 by A1, FUNCT_7:32;
  then reconsider G = G0 as PGraphMapping of F.x,(canGFDistinction(F,z)).x9;
  take G;
  thus thesis by A2, A3, A4;
end;
