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

theorem Th29:
  G2.allSG() c= G1.allSG() iff G2 is Subgraph of G1
proof
  hereby
    assume A1: G2.allSG() c= G1.allSG();
    G2 | _GraphSelectors in G2.allSG() by Th3;
    then A2: G2 | _GraphSelectors is Subgraph of G1 by A1, Th1;
    G2 == G2 | _GraphSelectors by GLIB_000:128;
    hence G2 is Subgraph of G1 by A2, GLIB_000:92;
  end;
  assume A3: G2 is Subgraph of G1;
  now
    let x be object;
    assume A4: x in G2.allSG();
    then reconsider H = x as _Graph;
    H is Subgraph of G2 by A4, Th1;
    then H is Subgraph of G1 by A3, GLIB_000:43;
    hence x in G1.allSG() by A4, Th1;
  end;
  hence thesis by TARSKI:def 3;
end;
