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Welling missed out on promotion to the National League after play-off final heartache at Woking on Sunday.
The Wings, who finished third in the league campaign, lost to the National League South's runners-up after conceding a goal three minutes before half-time.
It was a cracking strike that settled the tie but Welling had enough chances to have at least taken the contest into extra-time.
Welling settled quickest and Bradley Goldberg and Danny Mills both had attempts off target in the first 10 minutes.
One Woking free-kick apart, which keeper Dan Wilks flapped at, the home side never threatened the Wings goal in the early exchanges.
Welling could have easily got their noses in front in the opening half-hour. Mills caused Moussa Diarra plenty of problems in the air and, having won another header, he sent Goldberg clear but his right-foot shot came back off the inside of the post and into keeper Craig Ross' grateful arms.
Wings defender Josh Hill, in the team for the injured Anthony Acheampong, cleared off his own line after Reise Allassani brilliantly sliced open the visiting defence to set up Armani Little.
Woking went ahead following a needless, and rash, challenge by Thierry Audel on the edge of his own area. It presented Woking with a free-kick in an excellent position and Little brilliantly fired home to give his side a 42nd-minute lead.
Woking were denied a second before the hour mark. Hyde appeared offside from the first ball but when he was played in again moments later the flag stayed down and it needed a smart stop from Wilks to divert his effort behind for a corner.
Welling introduced Nassim L'Ghoul and Julien Anau - a switch that nearly paid immediate dividends. L'Ghoul advanced into the penalty area and appeared to be tripped by Jack Cook, but referee Andrew Kitchen turned down their appeals for a spot-kick.
Mills continued to carry Welling's biggest threat, his 20-yard shot was parried away by Ross but no-one was on hand to convert the rebound.
Welling had plenty of late pressure, although they lost some of their momentum as Woking managed to successfully wind the clock down - which included a lengthy delay for Ross to tie his laces up again.
Hyde forced Wilks into another save as Woking countered on the break and Welling - for all their possession - were struggling to create clear-cut chances.
Welling were unable to fashion a chance in five minutes of stoppage time with the referee almost already down the tunnel to avoid the inevitable pitch invasion when he blew the final whistle.
Woking: Ross, Cook, Casey, Ferdinand, Gerring, Diarra, Allassani (Durojaiye 87mins), Little, Hyde (Bradbury 90mins), Kretzschmar, Cadogan (Hodges 75mins). Subs not used: Collier, Mason.
Welling: Wilks, Gibbons, Braham-Barrett, Mendy (Annau 64mins), Hill, Audel, Kiernan, Ijaha, Mills, Goldberg (Bettamer 72mins), McCallum (L'Ghoul 64mins). Subs not used: Hajizadeh, Orlu.
Attendance: 4,865 (809 away).