
theorem Th55:
  for G being _Graph, v being Vertex of G
  st v is isolated holds {v} = G.reachableFrom(v)
proof
  let G be _Graph, v be Vertex of G;
  assume A1: v is isolated;
  for x being object holds x in {v} iff x in G.reachableFrom(v)
  proof
    let x be object;
    hereby
      assume x in {v};
      then x = v by TARSKI:def 1;
      hence x in G.reachableFrom(v) by GLIB_002:9;
    end;
    assume A2: x in G.reachableFrom(v);
    then reconsider v2 = x as Vertex of G;
    v = v2 by A1, A2, Th53;
    hence x in {v} by TARSKI:def 1;
  end;
  hence thesis by TARSKI:2;
end;
