
theorem Th101:
  for F being non empty Graph-yielding Function, x,z being Element of dom F
  st x <> z holds (the_Edges_of canGFDistinction(F,z)).x
    = (the_Edges_of canGFDistinction F).x
proof
  let F be non empty Graph-yielding Function, x,z be Element of dom F;
  assume A1: x <> z;
  reconsider x1 = x as Element of dom canGFDistinction(F) by Def25;
  reconsider x2 = x as Element of dom canGFDistinction(F,z) by Th95;
  thus (the_Edges_of canGFDistinction(F,z)).x
     = the_Edges_of(canGFDistinction(F,z).x2) by Def9
    .= the_Edges_of((canGFDistinction F).x1) by A1, FUNCT_7:32
    .= (the_Edges_of canGFDistinction F).x by Def9;
end;
