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Derbyshire 318 for 8 (Guest 116, Dal 106) trail Leicestershire 528 by 215 runs
The pair scored maiden first-class centuries as they passed Derbyshire's previous highest sixth-wicket stand of 212 by Garnet Lee and Stan Worthington against Essex at Chesterfield in 1932.
They added 227 in 73 overs with Guest scoring 116 and Dal 106 to lift Derbyshire from 56 for 5 to 313 for 8 although 65 is still needed to avoid following on.
The prospect of Derbyshire frustrating Leicestershire's bowlers looked unlikely when two wickets fell in the first seven overs of another hot morning.
Resuming on 41 for 3, a daunting 487 runs behind, Derbyshire lost Billy Godleman and Matt Critchley with only 15 added.
Godleman was drawn into playing at a ball from Chris Wright that moved just enough to take the outside edge and give Harry Swindells his fourth catch.
If there was some mitigation for the Derbyshire captain, there was none for Critchley, who launched a loose drive at Ben Mike and edged the ball into his middle stump.
At that stage, the chances of Derbyshire getting close to the follow-on target of 378 appeared slim but Guest and Dal showed the application required.
A delay caused by one of Dal's contact lenses falling out did not disturb their concentration as they wore down the Leicestershire attack.
Both showed patience and judgement to survive until lunch and they prospered in the afternoon with Guest cutting Mike for his seventh four to reach 50 off 98 balls.
Dal drove George Rhodes to the cover boundary to bring up his 50 from 122 balls and Guest swept the off-spinner for four to earn Derbyshire a batting point in the 68th over.
There was little assistance for Callum Parkinson's left-arm spin but he delayed taking the new ball until after tea when Guest survived a strong lbw shout against Wright.
His hundred, which came off 221 balls arrived with three fours in an over off former Derbyshire seamer Will Davis, and Dal followed by pulling Mike for his 15th four to reach three figures from 210 balls.
The stand ended when Dal miscued a pull and after Alex Thomson was lbw to Parkinson, Guest was caught behind cutting to still leave Derbyshire with work to do on the fourth day.
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Derbyshire captain cashes in in the follow-on to prevent against last-day dramas
Brooke Guest, Anuj Dal put on record stand to rescue Derbyshire
Pair score maiden first-class tons for Derbyshire's highest sixth-wicket partnership but hard work remains
Chris Wright has a day to remember as Leicestershire take control
Fast bowler claims 500th first-class wicket, scores career-best 87 as visitors take big lead in Derby