
theorem Th73:
  for F being non empty Graph-yielding Function
  holds F is vertex-disjoint edge-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 &
      (the_Edges_of F).x1 misses (the_Edges_of F).x2
proof
  let F be non empty Graph-yielding Function;
  thus F is vertex-disjoint edge-disjoint implies
    for x1,x2 being Element of dom F st x1 <> x2 holds
      (the_Vertices_of F).x1 misses (the_Vertices_of F).x2 &
      (the_Edges_of F).x1 misses (the_Edges_of F).x2 by Th70, Th71;
  assume A1: for x1,x2 being Element of dom F st x1 <> x2
    holds (the_Vertices_of F).x1 misses (the_Vertices_of F).x2 &
      (the_Edges_of F).x1 misses (the_Edges_of F).x2;
  then for x1,x2 being Element of dom F st x1 <> x2
    holds (the_Vertices_of F).x1 misses (the_Vertices_of F).x2;
  hence F is vertex-disjoint by Th70;
  for x1,x2 being Element of dom F st x1 <> x2
    holds (the_Edges_of F).x1 misses (the_Edges_of F).x2 by A1;
  hence F is edge-disjoint by Th71;
end;
