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 G1 being _Graph, G2 being Subgraph of G1, v1 being Vertex of G1,
  v2 being Vertex of G2 st v1 = v2 & v1 is isolated holds v2 is isolated
proof
  let G1 be _Graph, G2 be Subgraph of G1, v1 be Vertex of G1, v2 be Vertex of
  G2;
  assume v1 = v2 & v1 is isolated;
  then v1.edgesInOut() = {} & v2.edgesInOut() c= v1.edgesInOut() by Th78;
  hence thesis;
end;
