
theorem Th44: :: VAdjacent01
  for G being _Graph, S being non empty Subset of the_Vertices_of
  G for H being inducedSubgraph of G,S for u, v being Vertex of G, t, w being
  Vertex of H st u = t & v = w holds u,v are_adjacent iff t,w are_adjacent
proof
  let G be _Graph, S be non empty Subset of the_Vertices_of G;
  let H be inducedSubgraph of G,S;
  let u, v be Vertex of G, t, w be Vertex of H such that
A1: u = t and
A2: v = w;
    S = the_Vertices_of H by GLIB_000:def 37;
  hence u,v are_adjacent implies t,w are_adjacent by A1,A2,Th19;
  assume t,w are_adjacent;
  then consider e being object such that
A3: e Joins t,w,H;
  e Joins u,v,G by A1,A2,A3,GLIB_000:72;
  hence thesis;
end;
