Adding six unit lower bounds

Let $a\geq 1,\ b\geq 1,\ c\geq 1,\ d\geq 1,\ e\geq 1,\ f\geq 1$ . Prove that $a+b+c+d+e+f\geq 6$ .

Source: InternLM Lean-Workbook, record lean_workbook_9922 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.