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

theorem Th41:
  SubgraphRel(G) partially_orders G.allSG()
proof
  now
    let x be object;
    assume x in G.allSG();
    then reconsider H = x as Element of G.allSG();
    H is Subgraph of H by GLIB_000:40;
    hence [x,x] in SubgraphRel(G) by Def6;
  end;
  then A1: SubgraphRel(G) is_reflexive_in G.allSG() by RELAT_2:def 1;
  now
    let x,y be object;
    assume that
      A2: x in G.allSG() & y in G.allSG() and
      A3: [x,y] in SubgraphRel(G) & [y,x] in SubgraphRel(G);
    reconsider H1 = x, H2 = y as Element of G.allSG() by A2;
    H1 is Subgraph of H2 & H2 is Subgraph of H1 by A3, Def6;
    hence x = y by GLIB_000:87, GLIB_009:44;
  end;
  then A4: SubgraphRel(G) is_antisymmetric_in G.allSG()
    by RELAT_2:def 4;
  now
    let x,y,z be object;
    assume that
      A5: x in G.allSG() & y in G.allSG() & z in G.allSG()
      and A6: [x,y] in SubgraphRel(G) & [y,z] in SubgraphRel(G);
    reconsider H1 = x, H2 = y, H3 = z as Element of G.allSG() by A5;
    H1 is Subgraph of H2 & H2 is Subgraph of H3 by A6, Def6;
    then H1 is Subgraph of H3 by GLIB_000:43;
    hence [x,z] in SubgraphRel(G) by Def6;
  end;
  hence thesis by A1, A4, RELAT_2:def 8, ORDERS_1:def 8;
end;
