On quasi-invariant elements of a lattice

Let L be a lattice and G a group acting on L. An element x of L is said to be quasi-G-invariant if for every g is an element of G either x less than or equal to g(x) or x covers x boolean AND g(x). We prove that if L is an upper semimodular lattice and x is an element of L is quasi-G-invariant then there is a g is an element of G such that I covers x boolean AND g(x) and x boolean AND g(x) is G-invariant, or there is a g is an element of G such that x boolean OR g(x) covers x and x boolean OR g(x) is G-invariant.