
theorem Th17:
  for G being locally-finite _Graph holds G is Path-like iff
    G is Tree-like & for v being Vertex of G holds v.degree() <= 2
proof
  let G be locally-finite _Graph;
  hereby
    assume A1: G is Path-like;
    hence G is Tree-like;
    let v be Vertex of G;
    Segm v.degree() c= Segm 2 by A1;
    hence v.degree() <= 2 by NAT_1:39;
  end;
  assume A2: G is Tree-like;
  assume A3: for v being Vertex of G holds v.degree() <= 2;
  thus G is Tree-like by A2;
  let v be Vertex of G;
  Segm v.degree() c= Segm 2 by A3, NAT_1:39;
  hence v.degree() c= 2;
end;
