Let F1,F2,F3,...........Fn represent the sequence of Fibonacci elements. Let us define F to be the parent set of all Fibonacci elements. G and G′ are the subsets of F such that G is a given set of consecutive Fibonacci elements of finite order k and G′ is defined to be a shift on G of l degrees, where l ∈ N. Let R = min(r1,r2,....) denote the set of remainders obtained such that rn ∈ F. For a given G of order k, we show that a strategic mapping operator ϕ: (G × G) −→ R defined by §: ϕ(g ⊗g ′ h) = r, where (G × G) represents the Cartesian product and g, h ∈ G , g ′ ∈ G′ . The strategic map ϕ exists upto (l + 1)0 transition, with its limit as L Fn+(l+1) thereof. We consider a special introductory case of |G|, |G′ |=4 to illustrate the results and thereby proving the ”Fundamental Theorem of limit of a strategic map of Fibonacci sequence[Thomas heorem] and its consequences”.