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.edgesInOut() = G.edgesBetween(the_Vertices_of G,{v})
proof
  let G be _Graph, v be Vertex of G;
  thus v.edgesInOut() = v.edgesIn() \/ v.edgesOut()
    .= G.edgesDBetween(the_Vertices_of G,{v}) \/ v.edgesOut() by Th39
    .= G.edgesDBetween(the_Vertices_of G,{v}) \/
      G.edgesDBetween({v},the_Vertices_of G) by Th39
    .= G.edgesBetween(the_Vertices_of G,{v}) by Th150;
end;
