reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th65:
  G2.allSpanningSG() c= G1.allSpanningSG() iff G2 is spanning Subgraph of G1
proof
  hereby
    assume A1: G2.allSpanningSG() c= G1.allSpanningSG();
    G2 | _GraphSelectors in G2.allSpanningSG() by Th62;
    then G2 | _GraphSelectors is spanning Subgraph of G1 by A1, Th60;
    hence G2 is spanning Subgraph of G1 by GLIB_000:171, GLIB_000:128;
  end;
  assume A2: G2 is spanning Subgraph of G1;
  now
    let x be object;
    assume x in G2.allSpanningSG();
    then reconsider H = x as plain spanning Subgraph of G2 by Th60;
    the_Vertices_of H = the_Vertices_of G2 by GLIB_000:def 33
      .= the_Vertices_of G1 by A2, GLIB_000:def 33;
    then H is plain spanning Subgraph of G1
      by A2, GLIB_000:43, GLIB_000:def 33;
    hence x in G1.allSpanningSG() by Th60;
  end;
  hence thesis by TARSKI:def 3;
end;
