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

theorem Th63:
  for v,w being object holds [v,w] in E iff [v,w] DJoins v,w,createGraph(V,E)
proof
  let v,w be object;
  set G0 = createGraph(V,E);
  hereby
    assume A1: [v,w] in E;
    then A2: [v,w] in the_Edges_of G0;
    A3: v in V & w in V by A1, ZFMISC_1:87;
    A4: (the_Source_of G0).[v,w] = (pr1(V,V)|E).[v,w]
      .= pr1(V,V).[v,w] by A1, FUNCT_1:49
      .= pr1(V,V).(v,w) by BINOP_1:def 1
      .= v by A3, FUNCT_3:def 4;
    (the_Target_of G0).[v,w] = (pr2(V,V)|E).[v,w]
      .= pr2(V,V).[v,w] by A1, FUNCT_1:49
      .= pr2(V,V).(v,w) by BINOP_1:def 1
      .= w by A3, FUNCT_3:def 5;
    hence [v,w] DJoins v,w,G0 by A2, A4, GLIB_000:def 14;
  end;
  assume [v,w] DJoins v,w,G0;
  then [v,w] in the_Edges_of G0 by GLIB_000:def 14;
  hence [v,w] in E;
end;
