reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;

theorem
  for G being _finite _Graph holds G is Tree-like iff G is connected & G
  .order() = G.size() + 1
proof
  let G be _finite _Graph;
  thus G is Tree-like implies G is connected & G.order() = G.size()+1 by Th45;
  assume that
A1: G is connected and
A2: G.order() = G.size() + 1;
  now
    assume not G is acyclic;
    then consider W being Walk of G such that
A3: W is Cycle-like;
    set e = the Element of W.edges();
    set G2 = the removeEdge of G,e;
A4: W.edges() <> {} by A3,GLIB_001:136;
    then e in W.edges();
    then
A5: G2.order()=G.order() & G2.size()+1=G.size() by GLIB_000:52;
    G2 is connected by A1,A3,A4,Th4;
    then G2.size() + 1 + 1 <= G2.size() + 1 + 0 by A2,A5,Th39;
    hence contradiction by XREAL_1:6;
  end;
  hence thesis by A1;
end;
