Note on a change of coordinates for nonlinear SISO systems with singular points

In the paper titled `Asymptotic Output Tracking Through Singular Points for a class of Uncertain SISO Nonlinear Systems', a transformation T:Rⁿ×R⁺→R^n+β which takes (x,t)qq(Z,ζ,Φ) defined by Z₁ = h(x) Z₂ = L_fh(x)qqZ_α = L_f^α-1h(x) ζ_α+1 = a₀(x)+b₀(x)u ζ_α+2 = a₁(x,u)+b₀(x)u′+b₁(x,u)u ζ_α+3 = a₂(x,u,u′)+b₂(x,u,u′)u+2b₁(x,u)u′+ b₀( x)u″qqζ_α+β = a_β-1(x,u,qq,u^(β-2))+b_β-1(x,u, qq,u^{( β-2)})u+qq+b₀(x)u^(β-1) {Φ_i(x)}_{i = α+1}ⁿ was proposed in order to get rid of the singularity and being able to compute an admissible control u. There was also discussed that T was formed by at least n linearly independent functions as long as x≠x_s i.e., no singular points, (this was assumed when x = x_s). The main purpose of this paper is to present a proof where it is shown that T given by (1) provides us with a set of at least n linearly independent functions in case of having singular points.