、。
I'm terribly sorry, but it seems I've lost your umbrella.
Literal
Really sorry [concessive-のですが], I [topic-は] you [possessive-の] umbrella [object-を] lost [evidential-らしいのです].
申し訳ないのですが ('I'm sorry, but...') is a formal apology followed by のですが for softening. ~らしい ('it seems, apparently') hedges the claim — the speaker suspects rather than knows for certain. The layered politeness (apology + hedge + のです) is characteristic of Japanese indirectness.