The phrase "further solidify" is correct and can be used in written English.
It means to strengthen or reinforce something that is already strong or stable. Example: The new evidence presented in court further solidified the prosecutor's case against the defendant.
"Under his direction, Calvin Klein will further solidify its position as a leading global lifestyle brand".
His endorsement will further solidify Trump's credentials as a fervent opponent of illegal immigration.
The next round of auctions will further solidify the UK security of supply".
The government, however, responded by attempting to further solidify its authority.
These victories not only reflect strategic shifts by government forces but also could further solidify the country's divisions.
But the restrictions may only further solidify the changing world of banker pay, rather than shake up compensation practices.
Andrew Miller, CEO, Guardian Media Group (GMG), today announces a series of executive management changes that will further solidify the Guardian's position as a leading global news organisation.
The game is a chance for Strait, a third-round pick in the draft last April, to further solidify his place on the team.
If they do, William C. Ford Jr., Ford's chairman and the great-grandson of Henry Ford, will further solidify his reputation as the most adventurous executive in an industry not previously known for environmental stewardship.
Both Cashman and Expos General Manager Omar Minaya said yesterday that trade talks were still alive, but the sides had made no progress and a development yesterday may further solidify the teams' positions.
Of course, being associated with one of the few truly global clubs in football helps them capture market share in emerging markets and further solidify its presence in existing markets.
