From Summary: In the first section, the group of coordinates transformation is given and the natural bases $\bar B$ and $\bar B^*$ of tangent and cotangent spaces $T(GLH)^{(nk)}$ and $T^*(GLH)^{(nk)}$ are examined. In the second section, the solution of the variation problem of the integral of action for the extreme value of the fundamental function $F(x,y^1,\dots,y^k,p_1,\dots,p_k)$ is obtained. Here, the modified Liouville vectors $I_A(v,h)$ are applied.... The generalized Euler-Lagrange (E-L) equations in $(GLH)^{(nk)}$ reduce to the known (E-L) equations in generalized Lagrange spaces. In the third section, the generalizations of Craig-Synge covectors are given and some important theorems connected with this problem in $(GLH)^{(nk)}$ are proved.