Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph and g(x) and f(x) integer-valued functions defined on V(G) with g(x)less than or equal to f(x) for every x is an element of V(G). For a subgraph H of G and a factorization 9= \F-1, F-2,..., F-t\ of G, if \E(H)boolean AND E(F-i)\ = 1, 1 less than or equal to i less than or equal to t, then we say that F is orthogonal to H. It is proved that for an (mg(x) + k, mf(x)-k)-graph G, there exists a subgraph R of G such that for any subgraph H of G with \E(H)\ = k, R has a (g, f)-factorization orthogonal to H, where 1 less than or equal to k<m and g(x)greater than or equal to 1 or f(x)greater than or equal to 5 for every x is an element of V(G).