
theorem Th59:
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S
  st G is connected ex H being _Graph st S = {H}
proof
  let S be vertex-disjoint GraphUnionSet, G being GraphUnion of S;
  assume A1: G is connected;
  set v = the Vertex of G;
  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 H being _Graph such that
    A3: H in S & X = the_Vertices_of H by A2, GLIB_014:def 14;
  take H;
  now
    let x be object;
    assume x in the_Vertices_of G;
    then reconsider w = x as Vertex of G;
    consider W being Walk of G such that
      A4: W is_Walk_from v,w by A1, GLIB_002:def 1;
    consider H9 being Element of S such that
      A5: W is Walk of H9 by Th58;
    reconsider W9 = W as Walk of H9 by A5;
    W9 is_Walk_from v,w by A4, GLIB_001:19;
    then A6: v is Vertex of H9 & w is Vertex of H9 by GLIB_001:18;
    then the_Vertices_of H meets the_Vertices_of H9 by A2, A3, XBOOLE_0:3;
    then H = H9 by A3, Def18;
    hence x in the_Vertices_of H by A6;
  end;
  then A7: the_Vertices_of G c= the_Vertices_of H by TARSKI:def 3;
  now
    let Y be object;
    assume Y in S;
    then reconsider H9 = Y as Element of S;
    set w = the Vertex of H9;
    H9 is Subgraph of G by GLIB_014:21;
    then the_Vertices_of H9 c= the_Vertices_of G by GLIB_000:def 32;
    then w in the_Vertices_of G by TARSKI:def 3;
    then the_Vertices_of H meets the_Vertices_of H9 by A7, XBOOLE_0:3;
    then Y = H by A3, Def18;
    hence Y in {H} by TARSKI:def 1;
  end;
  hence thesis by TARSKI:def 3, ZFMISC_1:33;
end;
