reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;

theorem Th75:
  for G2, G1 for e,v1,v2 being object st e DJoins v1,v2,G1
    holds e DJoins v1,v2,G2 or not e in the_Edges_of G2
proof
  let G2, G1;
  let e,v1,v2 be object;
  assume A1: e DJoins v1,v2,G1;
  per cases;
  suppose not e in the_Edges_of G2;
    hence thesis;
  end;
  suppose A2: e in the_Edges_of G2;
    A3: e in the_Edges_of G1 &
      (the_Source_of G1).e = v1 &
      (the_Target_of G1).e = v2 by A1, GLIB_000:def 14;
    reconsider e1=e as set by TARSKI:1;
    (the_Source_of G2).e = v1 & (the_Target_of G2).e = v2 by A2, A3, Def9;
    hence thesis by GLIB_000:def 14;
  end;
end;
