なくて済む/ないで済む => get by without doing ~ N2
を[経「へ」]て => After A N1
Time, Situation
~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
出す => to begin to; to start to; to burst into; ... out (e.g. to jump out, to carry out)​ N4