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

theorem
  G.allComponents() c= G.allInducedSG() /\ G.allConnectedSG()
proof
  now
    let x be object;
    assume x in G.allComponents();
    then reconsider C = x as plain Component of G by Th189;
    set v = the Vertex of C;
    the_Vertices_of C c= the_Vertices_of G;
    then reconsider v as Vertex of G by TARSKI:def 3;
    set C9 = the plain inducedSubgraph of G,G.reachableFrom(v);
    the_Vertices_of C9 = G.reachableFrom(v) by GLIB_000:def 37
      .= the_Vertices_of C by GLIB_002:33;
    then C = C9 by GLIB_002:32, GLIB_009:44;
    then x in G.allInducedSG() & x in G.allConnectedSG() by Th124;
    hence x in G.allInducedSG() /\ G.allConnectedSG() by XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 3;
end;
