Proof of. Theorem 1.3.3: reduction to a grid We shall need some basic facts regarding polar decompositions. For A ∈ ¥(H), let |A| = √A∗A. If A is normal, then there exists a unitary operator U such that U|A| = |A|U = A. If A ∈ A C ¥(H) is not invertible, then U may not belong to A. If A ∈ A is invertible (but not necessarily normal), then there exists a unique unitary U ∈ A such that A = U|A|. The element U can be defined as U = A(A∗A)—1/2. It satisfies the important relation U|A| = |A∗|U. (3.1.1) Moreover, for invertible A, the condition A ∈ GL0(A) is equivalent to U ∈ GL0(A). An analogue of (3.1.1) holds for general bounded operators (if A is not invertible, then U is only a partial isometry), but we will not use it. Recall that diagP T = PTP + (1 — P )T (1 — P ) for T ∈ M2(A). The following simple lemma will be very helpful in establishing that certain elements belong to GL0(A ⊕ A).

Proof of. An employee, full or part-time, may be required to produce an acceptable certificate from a Medical Practitioner for any illness in excess of five (5) consecutive working days or at the discretion of the Employer certifying that he was unable to carry out his duties due to illness.