
theorem Th70:
  for F being non empty Graph-yielding Function holds F is vertex-disjoint iff
    for x1,x2 being Element of dom F st x1 <> x2
    holds (the_Vertices_of F).x1 misses (the_Vertices_of F).x2
proof
  let F be non empty Graph-yielding Function;
  hereby
    assume A1: F is vertex-disjoint;
    let x1, x2 be Element of dom F;
    assume A2: x1 <> x2;
    (the_Vertices_of F).x1 = the_Vertices_of(F.x1) &
      (the_Vertices_of F).x2 = the_Vertices_of(F.x2) by Def8;
    hence (the_Vertices_of F).x1 misses (the_Vertices_of F).x2 by A1, A2;
  end;
  assume A3: for x1,x2 being Element of dom F st x1 <> x2
    holds (the_Vertices_of F).x1 misses (the_Vertices_of F).x2;
  let x1,x2 being Element of dom F;
  assume A4: x1 <> x2;
  (the_Vertices_of F).x1 = the_Vertices_of(F.x1) &
    (the_Vertices_of F).x2 = the_Vertices_of(F.x2) by Def8;
  hence the_Vertices_of (F.x1) misses the_Vertices_of (F.x2) by A3, A4;
end;
