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Let $K$ be a convex set in a real Banach space, and let $A:K\to X^{\ast }$ be a strongly monotone operator. If $u,w\in K$ both satisfy the displayed variational inequality, testing with $v=w$ and $v=u$ respectively and adding yields $⟨Au-Aw,u-w⟩\le 0$. Strong monotonicity gives $m\Vert u-w{\Vert }^2\le 0$, hence $u=w$. This proves uniqueness whenever a solution exists.

The hymen is a variable fold or rim of tissue near the vaginal opening. Its shape and elasticity vary naturally, and it need not seal the opening. Appearance of the hymen cannot reliably establish a person's sexual history; it is an anatomical structure rather than a biological record of one event.^[ref]

The tonoplast is the cell membrane surrounding a plant vacuole. Its transport systems maintain differences in ion composition, pH and stored substances between the vacuolar interior and the cytoplasm. Proton pumping supplies an electrochemical gradient that can drive coupled transport. This makes the tonoplast central to cellular storage and osmosis, while distinguishing it from the outer plasma membrane surrounding the whole plant cell. ^[ref]

A uterine fibroid situated within the muscular wall of the uterus. The term locates the lesion anatomically; it does not alone specify symptoms or biological behavior.^[ref]

A uterine fibroid contacting or projecting into the uterine cavity beneath the endometrial surface. Cavity distortion distinguishes this location from a purely outward-growing lesion.^[ref]

A uterine fibroid growing toward or beyond the outer surface of the uterus. Its location makes pressure on surrounding structures more relevant than direct distortion of the uterine cavity.^[ref]

A uterine fibroid attached by a stalk. The term describes its attachment, while submucosal or subserosal describes its relation to uterine layers; these descriptors can therefore apply to the same lesion.^[ref]

The dense fibrous membrane separating superficial and deeper perineal structures; corresponds to the inferior layer of older triangular-ligament descriptions.

Historical anatomical model of perineal muscle and fascia; it should not be read as a universally uniform muscular sheet between two identical fascial layers.

Excess leukocytes in urine; this finding does not by itself prove bacterial infection or localise all lesions.

A salt of calcium and oxalate that can form crystals in urine and contribute to urinary stones. Crystal formation depends on solution conditions, so detecting crystals does not alone establish an obstructing stone.

A urinary cast containing red blood cells. Its tubular formation supports a renal origin of bleeding and is particularly associated with glomerular inflammation rather than bleeding confined to the lower urinary tract.^[ref]

A renal tubular cast containing white blood cells; a sign of intrarenal inflammation, not a diagnosis of one specific cause.

Historical description of transient functional obstruction attributed to muscular spasm. It must be distinguished from a fixed fibrotic urethral stricture and from other causes of urinary retention.

The median umbilical ligament is the fibrous remnant of the urachus running from the bladder apex toward the umbilicus. It is distinct from the paired medial umbilical ligaments associated with obliterated umbilical arteries. The similar names refer to different developmental remnants.

A structural diagram showing software classes or object interfaces, their attributes and operations, and relationships such as class inheritance and associations. A hollow triangle points toward a more general classifier; a dashed realization line distinguishes implementation of an interface. A plain association does not assert exclusive lifetime ownership of the referenced object.

Describe an interaction through participant lifelines and ordered messages. Time usually proceeds downward. Calls, replies and execution intervals show how participants cooperate in one scenario, with optional fragments for alternatives or repetition. The diagram emphasizes message ordering across participants.

Describe the possible states of a system and transitions triggered by events, optionally constrained by guards and accompanied by actions. The diagram emphasizes a component's lifecycle and permitted reactions. It can describe many interaction traces, while a sequence diagram focuses on the temporal ordering of messages in an interaction.

An ulcer arising with destruction of intestinal lymphoid tissue in typhoid fever. The classical larger lesion follows the longitudinal shape of a Peyer's patch.

Diarrhoea occurring during typhoid fever. The stool symptom does not measure the severity of systemic infection or exclude intestinal complications.