
theorem
  for C1 being Cycle-like _Graph, C2 being non acyclic Subgraph of C1
  holds C1 == C2
proof
  let C1 be Cycle-like _Graph, C2 be non acyclic Subgraph of C1;
  consider W2 being Walk of C2 such that
    A1: W2 is Cycle-like by GLIB_002:def 2;
  reconsider W1 = W2 as Walk of C1 by GLIB_001:167;
  W1 is Cycle-like by A1, GLIB_006:24;
  then W1.vertices() = the_Vertices_of C1 & W1.edges() = the_Edges_of C1
    by Th39;
  then W2.vertices() = the_Vertices_of C1 & W2.edges() = the_Edges_of C1
    by GLIB_001:98, GLIB_001:110;
  then A2: the_Vertices_of C1 c= the_Vertices_of C2 &
    the_Edges_of C1 c= the_Edges_of C2;
  C1 is Subgraph of C1 by GLIB_000:40;
  then C1 is Subgraph of C2 by A2, GLIB_000:44;
  hence thesis by GLIB_000:87;
end;
