theorem
  for G being _Graph, v being Vertex of G
  holds v is isolated iff v.degree() = 0
proof
  let G be _Graph, v be Vertex of G;
  hereby
    assume v is isolated;
    then v.inDegree() = 0 & v.outDegree() = 0 by Th156;
    hence v.degree() = card(0+0) by CARD_2:38
      .= 0;
  end;
  assume A1: v.degree() = 0;
  v.inDegree() c= v.degree() & v.outDegree() c= v.degree() by CARD_2:94;
  then v.inDegree() = 0 & v.outDegree() = 0 by A1, XBOOLE_1:3;
  hence thesis by Th156;
end;
