
theorem
  for G being Tree-like _Graph, H being spanning Subgraph of G
  st H is connected holds G == H
proof
  let G be Tree-like _Graph, H be spanning Subgraph of G;
  assume A1: H is connected;
  A2: G is Subgraph of G by GLIB_000:40;
  A3: the_Vertices_of G = the_Vertices_of H by GLIB_000:def 33;
  the_Edges_of G c= the_Edges_of H
  proof
    assume not the_Edges_of G c= the_Edges_of H;
    then the_Edges_of G \ the_Edges_of H <> {} by XBOOLE_1:37;
    then consider e being object such that
      A4: e in the_Edges_of G \ the_Edges_of H by XBOOLE_0:def 1;
    set v = (the_Source_of G).e, w = (the_Target_of G).e;
    A5: e in the_Edges_of G & not e in the_Edges_of H by A4, XBOOLE_0:def 5;
    then A6: e Joins v,w,G by GLIB_000:def 13;
    then reconsider v, w as Vertex of G by GLIB_000:13;
    reconsider v1 = v, w1 = w as Vertex of H by A3;
    consider W9 being Walk of H such that
      A7: W9 is_Walk_from v1, w1 by A1, GLIB_002:def 1;
    set P9 = the Path of W9;
    reconsider P = P9 as Path of G by GLIB_001:167, GLIB_001:176;
    P9 is_Walk_from v1,w1 by A7, GLIB_001:160;
    then A8: P is_Walk_from v,w by GLIB_001:19;
    G.walkOf(v,e,w) = G.pathBetween(v,w) by A6, GLIB_008:29
      .= P9 by A8, HELLY:def 2;
    then P9.edges() = G.walkOf(v,e,w).edges() by GLIB_001:110
      .= {e} by A6, GLIB_001:108;
    then e in P9.edges() by TARSKI:def 1;
    hence contradiction by A5;
  end;
  then G is Subgraph of H by A2, A3, GLIB_000:44;
  hence thesis by GLIB_000:87;
end;
