なくて済む/ないで済む => get by without doing ~ N2
を[経「へ」]て => After A N1
Time, Situation
~助詞+数量      => Corresponds to the verb...of quantity N5
Quantity
~をたて(~を経て)      => through; after; via N1
ずに済む => get by without doing ~ N2
て済む => sufficient by; no problem to; resolve by ~ N3