David Taylor-Smith and Ian Horseman Sewell, who quit their roles at G4S on Friday, are not expected to receive any pay-offs outside of notice entitlements within their contracts.
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Tory Eurosceptic Jacob Rees-Mogg played down the significance of the deal struck by the Government to win over pro-EU rebels.
The Tanaiste issued the warning to the UK ahead of a visit to Dublin by European Commission President Jean-Claude Juncker.
Around 9,500 people gathered at the neolithic World Heritage Site to attend the overnight celebration of the longest day of the year.