Simplified existence theorems on all fractional [a,b][a,b]-factors

Let GG be a graph with order nn and let g,f:V(G)→Ng,f:V(G)→N such that g(v)≤f(v)g(v)≤f(v) for all v∈V(G)v∈V(G). We say that GG has all fractional (g,f)(g,f)-factors if GG has a fractional pp-factor for every p:V(G)→Np:V(G)→N such that g(v)≤p(v)≤f(v)g(v)≤p(v)≤f(v) for every v∈V(G)v∈V(G). Let a