(Transformation theorem) For every $\epsilon>0$ there exists $\delta=\delta(n,\epsilon)>0$ such that if ${\rm Ric}_{M^{n}}\geq-(n-1)\delta^{2}$ and $u:B_{2}(p)\to\mathds{R}^{k}$ is a $\delta^{2}$-splitting map, then for each $x\in B_{1}(p)$ and $r\geq s^{\delta}_{x}$ there exists a lower triangular matrix $A=A(x,r)$ such that $A\circ u:B_{r}(x)\to\mathds{R}^{k}$ is a $\epsilon$-splitting map.