
theorem Th65:
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S
  holds G.componentSet()
    = union the set of all H.componentSet() where H is Element of S
proof
  let S be vertex-disjoint GraphUnionSet, G be GraphUnion of S;
  now
    let x be object;
    hereby
      assume x in G.componentSet();
      then consider v being Vertex of G such that
        A1: x = G.reachableFrom(v) by GLIB_002:def 8;
      the_Vertices_of G = union the_Vertices_of S by GLIB_014:def 25;
      then consider X being set such that
        A2: v in X & X in the_Vertices_of S by TARSKI:def 4;
      consider H9 being _Graph such that
        A3: H9 in S & X = the_Vertices_of H9 by A2, GLIB_014:def 14;
      reconsider w = v as Vertex of H9 by A2, A3;
      H9.reachableFrom(w) = x by A1, A3, Th64;
      then A4: x in H9.componentSet() by GLIB_002:def 8;
      H9.componentSet() in the set of all H.componentSet()
        where H is Element of S by A3;
      hence x in union the set of all H.componentSet() where H is Element of S
        by A4, TARSKI:def 4;
    end;
    assume x in union the set of all H.componentSet() where H is Element of S;
    then consider X being set such that
      A5: x in X & X in the set of all H.componentSet() where H is Element of S
      by TARSKI:def 4;
    consider H being Element of S such that
      A6: X = H.componentSet() by A5;
    consider w being Vertex of H such that
      A7: x = H.reachableFrom(w) by A5, A6, GLIB_002:def 8;
    H is Subgraph of G by GLIB_014:21;
    then the_Vertices_of H c= the_Vertices_of G by GLIB_000:def 32;
    then reconsider v = w as Vertex of G by TARSKI:def 3;
    x = G.reachableFrom(v) by A7, Th64;
    hence x in G.componentSet() by GLIB_002:def 8;
  end;
  hence thesis by TARSKI:2;
end;
