reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem
  for G2, v, V for G1 being addAdjVertexAll of G2, v, V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 &
    ex G3 being Component of G2 st the_Vertices_of G3 misses V
  holds G1 is non connected
proof
  let G2, v, V;
  let G1 be addAdjVertexAll of G2, v, V;
  assume that
    A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 and
    A2: ex G3 being Component of G2 st the_Vertices_of G3 misses V;
  ex G3 being Component of G2 st for w being Vertex of G3 holds not w in V
  proof
    consider G3 being Component of G2 such that
      A3: the_Vertices_of G3 misses V by A2;
    take G3;
    let w be Vertex of G3;
    the_Vertices_of G3 /\ V = {} by A3, XBOOLE_0:def 7;
    hence thesis by Lm7;
  end;
  hence thesis by A1, Th73;
end;
