reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th74:
  for X being set holds E|X = createGraph(V,E).edgesOutOf(X)
proof
  let X be set;
  set G = createGraph(V,E);
  now
    let z be object;
    hereby
      assume A1: z in E|X;
      then consider x,y being object such that
        A2: z = [x,y] by RELAT_1:def 1;
      A3: x in X & [x,y] in E by A1, A2, RELAT_1:def 11;
      then [x,y] DJoins x,y,G by Th63;
      then [x,y] in the_Edges_of G & (the_Source_of G).[x,y] = x
        by GLIB_000:def 14;
      hence z in G.edgesOutOf(X) by A2, A3, GLIB_000:def 27;
    end;
    set x = (the_Source_of G).z, y = (the_Target_of G).z;
    assume z in G.edgesOutOf(X);
    then A4: z in the_Edges_of G & x in X by GLIB_000:def 27;
    then z DJoins x,y,G by GLIB_000:def 14;
    then z = [x,y] by Th64;
    hence z in E|X by A4, RELAT_1:def 11;
  end;
  hence thesis by TARSKI:2;
end;
