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, X,Y being set holds G2
  .edgesBetween(X,Y) c= G1.edgesBetween(X,Y) & G2.edgesDBetween(X,Y) c= G1
  .edgesDBetween(X,Y)
proof
  let G1 be _Graph, G2 be Subgraph of G1, X,Y be set;
  now
    let x be object;
    assume x in G2.edgesBetween(X,Y);
    then x SJoins X,Y,G2 by Def30;
    then x SJoins X,Y,G1 by Th72;
    hence x in G1.edgesBetween(X,Y) by Def30;
  end;
  hence G2.edgesBetween(X,Y) c= G1.edgesBetween(X,Y);
  now
    let x be object;
    assume x in G2.edgesDBetween(X,Y);
    then x DSJoins X,Y,G2 by Def31;
    then x DSJoins X,Y,G1 by Th72;
    hence x in G1.edgesDBetween(X,Y) by Def31;
  end;
  hence thesis;
end;
