Bezout's Theorem and ideals of terminal forms

Let k   be any field, k[X1,…,Xn]k[X1,…,Xn] be the polynomial ring of n variables over k  . For any f=f0+f1+⋯+fr∈k[X1,…,Xn]f=f0+f1+⋯+fr∈k[X1,…,Xn] where each fifi is a homogeneous polynomial of degree i   and fr≠0fr≠0, define tm(f)=frtm(f)=fr. If I   is an ideal in k[X1,…,Xn]k[X1,…,Xn], define tm(I)tm(I) to be 〈tm(f):f∈I⧹{0}〉〈tm(f):f∈I⧹{0}〉, the ideal generated by the terminal forms tm(f)tm(f). Using Bezout's Theorem and Macaulay's Theorem, we will establish the following. If f,g∈k[X1,X2]f,g∈k[X1,X2] satisfying that gcd{f,g}=gcd{tm(f),tm(g)}=1gcd{f,g}=gcd{tm(f),tm(g)}=1 and I=〈f,g〉I=〈f,g〉, then tm(I)=〈tm(f),tm(g)〉tm(I)=〈tm(f),tm(g)〉. Actually the above result is equivalent to Bezout's Theorem, which sheds another perspective of Bezout's Theorem. These results are valid in k[X1,…,Xn]k[X1,…,Xn] also.