Mappings sharing a value on finite-dimensional spaces (Q1855011)

Let \(f\) and \(g\) be \(C^1\) maps between \(\mathbb{R}^n\) and \(\mathbb{R}^m\) with \(m\leq n\). Assuming that \(f\) is proper and has at least one zero, the author states a sufficient condition ensuring that the map \(f-g\) also has a zero. The proof is based on elementary arguments, but it is obscured by the careless writing (a sample of which is given by ``the product of Banach spaces \([0,1]\times X_1\) and \(X_2\)'' on page 394). It also seems that the statement of assumption (iii) should involve ``\(f(x)\neq tg(x)\)'' instead of ``\(f(x)\neq g(x)\)''.