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

theorem Th130:
  G2.allConnectedSG() c= G1.allConnectedSG() implies G2 is Subgraph of G1
proof
  assume A1: G2.allConnectedSG() c= G1.allConnectedSG();
  now
    let x be object;
    assume x in the_Vertices_of G2;
    then reconsider v = x as Vertex of G2;
    createGraph(v) in G2.allConnectedSG() by Th127;
    then createGraph(v) is Subgraph of G1 by A1, Th124;
    then the_Vertices_of createGraph(v) c= the_Vertices_of G1
      by GLIB_000:def 32;
    hence x in the_Vertices_of G1 by ZFMISC_1:31;
  end;
  then A2: the_Vertices_of G2 c= the_Vertices_of G1 by TARSKI:def 3;
  A3: the_Edges_of G2 c= the_Edges_of G1
  proof
    per cases;
    suppose G2 is edgeless;
      then the_Edges_of G2 = {};
      hence thesis by XBOOLE_1:2;
    end;
    suppose G2 is non edgeless;
      then reconsider G3 = G2 as non edgeless _Graph;
      now
        let x be object;
        assume x in the_Edges_of G3;
        then reconsider e = x as Edge of G3;
        createGraph(e) in G2.allConnectedSG() by Th128;
        then createGraph(e) is Subgraph of G1 by A1, Th124;
        then the_Edges_of createGraph(e) c= the_Edges_of G1 by GLIB_000:def 32;
        then {e} c= the_Edges_of G1 by Th13;
        hence x in the_Edges_of G1 by ZFMISC_1:31;
      end;
      hence thesis by TARSKI:def 3;
    end;
  end;
  now
    let e0 be set;
    assume A4: e0 in the_Edges_of G2;
    then reconsider G3 = G2 as non edgeless _Graph;
    reconsider e = e0 as Edge of G3 by A4;
    createGraph(e) in G2.allConnectedSG() by Th128;
    then A5: createGraph(e) is Subgraph of G1 by A1, Th124;
    the_Edges_of createGraph(e) = {e} by Th13;
    then A6: e0 in the_Edges_of createGraph(e) by TARSKI:def 1;
    then (the_Source_of createGraph(e)).e0 = (the_Source_of G1).e0 &
      (the_Target_of createGraph(e)).e0 = (the_Target_of G1).e0
      by A5, GLIB_000:def 32;
    hence (the_Source_of G2).e0 = (the_Source_of G1).e0 &
      (the_Target_of G2).e0 = (the_Target_of G1).e0 by A6, GLIB_000:def 32;
  end;
  hence thesis by A2, A3, GLIB_000:def 32;
end;
