on condition that 以...为条件
on the supposition that 以...为假设
on the ground(s) that 原因是；根据
with the exception that 除了
in spite of the fact that 尽管事实是
on the understanding that 条件是
on the assumption that 基本...的假设
He resigned on the ground(s) that his health was failing.
They reached a merger agreement on the understanding that both parties would be committed to expanding the market share of the company in Europe.