reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

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;
