
theorem
  for F being non empty Graph-yielding Function, z being Element of dom F
  holds the_Vertices_of canGFDistinction(F,z)
    = (the_Vertices_of canGFDistinction F) +* (z, the_Vertices_of (F.z))
proof
  let F be non empty Graph-yielding Function, z be Element of dom F;
  A1: dom the_Vertices_of canGFDistinction(F,z)
     = dom canGFDistinction(F,z) by Def4
    .= dom canGFDistinction(F) by FUNCT_7:30;
  then A2: dom the_Vertices_of canGFDistinction(F,z)
     = dom the_Vertices_of canGFDistinction(F) by Def4
    .= dom((the_Vertices_of canGFDistinction F) +* (z, the_Vertices_of (F.z)))
      by FUNCT_7:30;
  now
    let x be object;
    assume x in dom the_Vertices_of canGFDistinction(F,z);
    then reconsider x0 = x as Element of dom F by A1, Def25;
    per cases;
    suppose A3: x = z;
      z in dom F;
      then z in dom canGFDistinction(F) by Def25;
      then A4: z in dom the_Vertices_of canGFDistinction F by Def4;
      thus (the_Vertices_of canGFDistinction(F,z)).x
         = (the_Vertices_of F).z by A3, Th97
        .= the_Vertices_of (F.z) by Def8
        .= ((the_Vertices_of canGFDistinction F)+*(z,the_Vertices_of (F.z))).x
          by A3, A4, FUNCT_7:31;
    end;
    suppose A5: x <> z;
      hence (the_Vertices_of canGFDistinction(F,z)).x
         = (the_Vertices_of canGFDistinction F).x0 by Th98
        .= ((the_Vertices_of canGFDistinction F)+*(z,the_Vertices_of (F.z))).x
          by A5, FUNCT_7:32;
    end;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
