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, X being set, v being Vertex of G st v is isolated
  holds G.edgesBetween(X \ {v}) = G.edgesBetween(X)
proof
  let G be _Graph, X be set, v be Vertex of G;
  assume A1: v is isolated;
  thus G.edgesBetween(X \ {v}) = G.edgesBetween(X) \ v.edgesInOut() by Th107
    .= G.edgesBetween(X) by A1;
end;
