
theorem Th49: :: Adjacent00
  for G being _Graph, S being set for u being Vertex of G holds u
  in G.AdjacentSet(S) iff not u in S & ex v being Vertex of G st v in S & u,v
  are_adjacent
proof
  let G be _Graph, S be set;
  let u be Vertex of G;
  hereby
    assume u in G.AdjacentSet(S);
    then
    ex t being Vertex of G st u=t &( not t in S)& ex v being Vertex of G
    st v in S & t,v are_adjacent;
    hence not u in S & ex v being Vertex of G st v in S & u,v are_adjacent;
  end;
  assume that
A1: not u in S and
A2: ex v being Vertex of G st v in S & u,v are_adjacent;
  thus thesis by A1,A2;
end;
