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 v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2 & G1 == G2
holds (v1 is isolated implies v2 is isolated) & (v1 is endvertex implies v2 is
  endvertex)
proof
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume
A1: v1 = v2 & G1 == G2;
  hence v1 is isolated implies v2 is isolated by Th96;
  assume v1 is endvertex;
  then consider e being object such that
A2: v1.edgesInOut() = {e} & not e Joins v1,v1,G1;
  v2.edgesInOut() = {e} & not e Joins v2,v2,G2 by A1,A2,Th96;
  hence thesis;
end;
