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

theorem Th43:
  SubgraphRel(H) = (SubgraphRel G) |_2 H.allSG()
proof
  A1: H.allSG() c= G.allSG() by Th29;
  reconsider R = (SubgraphRel G) |_2 H.allSG()
    as Relation of H.allSG();
  now
    let H1, H2 be Element of H.allSG();
    hereby
      assume [H1,H2] in SubgraphRel(H);
      then A2: H1 is Subgraph of H2 by Def6;
      H1 in G.allSG() & H2 in G.allSG() by A1, TARSKI:def 3;
      then [H1,H2] in SubgraphRel(G) by A2, Def6;
      hence [H1,H2] in R by MMLQUER2:4;
    end;
    assume [H1,H2] in R;
    then A3: [H1,H2] in SubgraphRel(G) by MMLQUER2:4;
    then H1 in field SubgraphRel(G) & H2 in field SubgraphRel(G)
      by RELAT_1:15;
    then H1 in G.allSG() & H2 in G.allSG() by Th40;
    then H1 is Subgraph of H2 by A3, Def6;
    hence [H1,H2] in SubgraphRel(H) by Def6;
  end;
  hence thesis by RELSET_1:def 2;
end;
