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 holds G.edgesBetween(X,X) = G.edgesBetween(X)
proof
  let G be _Graph, X be set;
  now
    let e be object;
    hereby
      assume e in G.edgesBetween(X,X);
      then e SJoins X,X,G by Def30;
      then e in the_Edges_of G & (the_Source_of G).e in X &
        (the_Target_of G).e in X;
      hence e in G.edgesBetween(X) by Lm5;
    end;
    assume e in G.edgesBetween(X);
    then e in the_Edges_of G & (the_Source_of G).e in X &
      (the_Target_of G).e in X by Lm5;
    then e SJoins X,X,G;
    hence e in G.edgesBetween(X,X) by Def30;
  end;
  hence thesis by TARSKI:2;
end;
